This java applet is a simulation that demonstrates scalar waves in two dimensions. Wave motion crops up in many different areas in physics; water waves, sound, and light are three examples.

When the applet starts up you will see a white circle (called the "source") emitting circular waves. The light areas are positive and the dark areas are negative. So, if you prefer to think of the waves as sound waves, the light areas would be areas of high pressure, and the dark areas would be low pressure. The source might be a speaker of some sort.

The first thing to do when starting up the applet is to adjust the settings for your computer. Slide the "Resolution" slider to the right as far as you can without slowing things down too much. Or you could slide it to the left if things are already too slow. Another thing to try, if the simulation is too slow, is to slide the "Simulation Speed" button to the right. That won't improve the framerate (it will slow it down slightly, in fact) but it will get the waves moving faster. Or you could make the window smaller. The applet runs slower when the window is large.

Now you can start playing with the applet. You can drag the source around wherever you want. Also you can create new waves (areas of high pressure) by clicking anywhere. There is a popup menu that controls what the mouse does. By default it is set to "Mouse = Edit Wave". If you change it to "Edit Walls", then you can use the mouse to put obstacles in the wave's path.

The Setup popup can be used to view some interesting pre-defined experiments. Once an experiment is selected, you may modify it all you want. The choices are:

• Single Source: this is just a single source emitting circular waves.
• Two Sources: this is just two sources emitting circular waves, creating an interference pattern between them.
• Four Sources.
• Single Slit: this demonstrates diffraction of waves travelling through a slit.
• Double Slit: this demonstrates diffraction of waves travelling through a double slit.
• Triple Slit.
• Obstacle: this demonstrates diffraction of waves travelling around an obstacle.
• Half Plane: this demonstrates diffraction of waves around the edge of a plane.
• Dipole Source: this demonstrates an acoustic dipole source consisting of two sources out of phase.
• Lateral Quadrupole: this demonstrates an acoustic quadrupole source consisting of four sources arranged in a square.
• Linear Quadrupole: this demonstrates an acoustic quadrupole source consisting of four sources arranged in a line.
• Hexapole.
• Octupole.
• 12-Pole.
• Plane Wave: this demonstrates a simple plane wave source.
• Intersecting Planes: this demonstrates two plane waves intersecting at right angles.
• Phased Array 1: this is a group of point sources arranged in a line, where the relative phases of each point is different. This causes the radiation to be pointed downwards. The angle can be adjusted with the Phase Difference slider.
• Phased Array 2: this is a group of point sources arranged in a line, where the relative phases of each source causes the radiation to be directed at a single point. The location of the point can be adjusted with the Phase Difference slider.
• Phased Array 3: this is a group of point sources arranged in a line, where the relative phases of each source causes the radiation to act as if it is coming from a point source to the left of the screen. The location of the virtual point source can be adjusted with the Phase Difference slider.
• Doppler Effect 1: this shows a moving source, thereby demonstrating the Doppler effect.
• Doppler Effect 2: this shows waves being reflected by a moving obstacle. The horizontal obstacle on the right is moving up and down. (The divider in the middle is just there to make the effect more clear.) When the obstacle is moving up, the reflected waves have a higher frequency than the source. When it's moving down, the reflections have a lower frequency.
• Sonic Boom: this shows a source moving faster than the speed of wave propagation, creating a shock wave behind it.
• Big 1x1 Mode: this creates a small box with a standing wave in its normal mode of oscillation. The inside of the box changes color with a simple time dependence with no left-right or up-down motion.
• 1x1 Modes: this creates several small boxes of different sizes in their fundamental modes. If you cut out the right side of one of the boxes and turn up the brightness you can see waves coming out of the box at its resonant frequency.
• 1xN Modes: this creates several small boxes in other normal modes.
• NxN Modes: this creates several small boxes in other normal modes.
• 1xN Mode Combos: this creates several small boxes, each of which has a combination of two random 1xN modes.
• NxN Mode Combos: this creates several small boxes, each of which has a combination of two random NxN modes.
• 0x1 Acoustic Modes: this creates several small boxes of different sizes in their fundamental modes. The Fixed Edges checkbox is off, which causes the waves to act like acoustic waves.
• 0xN Acoustic Modes: this creates several small boxes in other acoustic normal modes. The mode frequencies are all multiples of the fundamental, so you will see all the modes sync up periodically. (This is not the case with the 1xN Modes example above.)
• NxN Acoustic Modes: this creates several small boxes in other acoustic normal modes.
• Coupled Cavities: this creates pairs of boxes with a small interconnection between them. This causes the oscillation energy to move back and forth between the two boxes.
• Beats: this creates two sources close together with different frequencies. Because the frequencies are close but not exactly the same, you will see black lines of interference or "beats".
• Slow Medium: in this demonstration, the area below the blue line has a different refractive index, so that waves move slower through that area. As a result, waves hitting the the blue region will be partially reflected and partially transmitted. Waves travel through the blue region at half the speed as they travel through the black region, so the blue region has a refractive index of 2. As a comparison, most common types of glass have a refractive index anywhere from 1.46 to 1.96.
• Refraction: this creates a blue region similar to the last setup, but shows short pulses hitting it at an angle so you can see the waves being reflected and refracted.
• Internal Reflection: this creates a blue region similar to the last setup, but shows short pulses hitting it at an angle from inside the blue region. The angle is such that none of the main part of the wave is transmitted; this is called total internal reflection. You will see some activity in the upper area; this is partly because the top part of the wave is rounded instead of being a plane, so that it hits the interface at a different angle (it goes up from the source instead of diagonally). This part is transmitted, but the plane part going diagonally is reflected. But, even for the part of the wave that is reflected, you will a portion of the wave travelling along the interface between the blue and black area; but it will not propagate into the black area.
• Anti-Reflective Coating: this creates a blue region similar to the last setup, but with an anti-reflective coating which eliminates reflections. (There will be some initial reflections when the source is first turned on because of high-frequency components in the initial wavefront.) If you increase the frequency of the source, you will see reflections because the wavelength of the source no longer matches the thickness of the coating.
• Zone Plate (Even): This creates a zone plate, which uses diffraction to focus light.
• Zone Plate (Odd): This creates another zone plate which is similar to the previous one, but has opaque areas made transparent and vice versa. It also focuses light.
• Circle: This creates a circular area with a source at the center. Pulses will travel outward and will then be reflected back to the center.
• Ellipse: This creates an elliptical area with a source at one focus. Pulses will converge at the other focus.
• Resonant Cavities 1: This creates a series of rectangular cavities being driven by a plane wave from above. As you change the frequency you will see the response of each cavity change. Each cavity has a different resonant frequency so it will respond differently. After changing the frequency you may want to wait a bit for things to settle down (or turn the simulation speed way up).
• Resonant Cavities 2: This creates a series of smaller rectangular cavities.
• Room Resonance: This shows acousting standing waves in a series of rooms being driven by the same frequency, but at different positions. The brightness is turned way down so you only see waves in the rooms that resonate. As you can see, three of the rooms resonate but the fourth does not, because the source is not located in the right place (there aren't any modes with the right frequency that have antinodes at the source location). By varying the frequency you can see different resonance behavior. (You may want to turn the simulation speed up so you get faster results as you experiment.)
• Waveguides 1: This creates a series of waveguides of different widths. Narrower waveguides, like at the left end of the screen, have higher cutoff frequencies; the leftmost waveguide has a cutoff frequency that is higher than the source, so there is no wave motion in it. You can fix this by turning the Source Frequency slider up.

  Notice that the waves seem to be moving faster in thinner waveguides. They appear to be moving faster than waves normally move in the applet. This is because the phase velocity is faster in thinner waveguides; but the signal velocity is actually slower than normal, as you can verify by clicking the Clear Waves button and watching the wave move down the guide for the first time.

  Since the waveguides are being driven by a plane wave, only the TE01 mode is present. (See the waveguide applet for another way to view waveguide modes.)

• Waveguides 2: This is just the same set of waveguides with a lower frequency, showing that more of the guides are driven below cutoff.
• Waveguides 3: This is a set of identical waveguides being driven by small holes at different locations. This causes different sets of modes to be excited in different proportions. When the guide is being driven near the center, the TE01 mode is dominant, but when it is driven near the edge, the TE02 mode is more prevalent. The frequency is low enough so that all other modes are cut off. You can fix this by turning the frequency up. By turning the frequency down, you can cut off the TE02 mode as well.
• Waveguides 4: This is a set of acoustic waveguides being driven at various locations.
• Waveguides 5: This is a set of identical waveguides with various modes present. The first waveguide contains the TE01 mode; the second contains the TE02 mode; the third contains both; the fourth contains the TE03 mode; the fifth contains TE01 and TE03; the sixth contains TE02 and TE03. (There may not be room on your screen for all these modes if your resolution is not set high enough.)

  Notice that the higher modes (TE02 and especially TE03) seem to be moving faster. This is because the phase velocity of TE02 and TE03 is greater than that of TE01. Their signal velocities are slower, though, which is why it takes the TE03 wave so long to make it down to the end of the waveguide. Also if you turn off the source (by setting the source popup to "No Sources") it will take quite a while for the TE03 mode to stop.

• Parabolic Mirror 1: This shows a parabola with a source at the focus. The parabola direct the waves upward as plane waves (except at the edges where the waves don't look planar; if we extended the parabola further it would fix this).
• Parabolic Mirror 2: This shows a parabola with plane waves coming from above. They converge at the focus.
• Sound Duct: This shows a duct with sound waves travelling through it. When they get to the end, they are partially reflected, even through there is nothing there for them to bounce off of. This shows that waves are reflected by any discontinuity, not just by walls.
• Baffled Piston: This shows the sound radiation from a baffled piston, which is a simple model of a boxed loudspeaker.
• Low-Pass Filter 1, 2: This shows an acoustic low-pass filter. Low-frequency waves travel through it more easily than high-frequency waves, as you can verify by comparing Low-Pass Filter 1 with Low-Pass Filter 2. (These two setups are the same except for the frequency.) However there are a few higher frequencies which will pass easily. If you follow the link you can take a look at the frequency response curve.
• High-Pass Filter 1, 2: This shows an acoustic high-pass filter. High-frequency waves travel through it more easily than low-frequency waves, as you can verify by comparing High-Pass Filter 1 with High-Pass Filter 2.
• Band-Stop Filter 1, 2, 3: This shows an acoustic band-stop filter, which blocks out a range of frequencies. There are three versions of this setup; one at a low frequency, one high, and one at the blocked frequency.
• Planar convex lens: This shows a lens made out of a glasslike material. It focuses plane waves. Unfortunately the lens is pretty small compared to the wavelength of light so it won't focus the light as well as it would in real life. This lens is only a dozen or so wavelengths wide. The range of visible light wavelengths is 400 to 700 nanometers, so obviously a real lens is much larger compared to a wavelength and so will focus better without running into diffraction effects.
• Biconvex lens: This shows another lens. It takes line coming from a point source and focuses it at another point.
• Planar Concave Lens: This shows a lens that takes plane waves and spreads them out.
• Circular Prism: This shows a round prism made out of a dense material.
• Right-Angle Prism: This shows a prism that takes waves travelling down and points them to the right.
• Porro Prism: This shows a prism that takes waves travelling down and points them up. Obviously in real life it would do this at light speed.
• Scattering: This shows a plane wave being scattered by a point particle.
• Lloyd's Mirror: This shows an interferometer which consists of a point source close to a mirror (at the bottom of the window). The waves coming from the source interfere with the waves coming from its mirror image.
• Temperature Gradient 1: This shows refraction of a wave due to a temperature gradient. The blue area represents cool air, where sound waves move more slowly. This causes the waves to bend downwards.
• Temperature Gradient 2: This shows refraction of a wave due to a different type of temperature gradient. The blue area represents cool air, where sound waves move more slowly. This causes the waves to bend upwards. A similar effect is responsible for mirages.
• Dispersion: This demonstrates dispersion, which is an effect that causes waves to move at different speeds depending on their frequency. This effect is unavoidable because of the finite differencing method used to design this applet. Note that the low-frequency waves on the left move faster than the high-frequency waves on the right. The waves are moving through a slow medium to illustrate the effect better, but dispersion occurs even without a medium in this applet.

  Usually the effect is not noticeable, unless you are using sources of two different frequencies. However, if you use the mouse to edit the wave function (using the Edit Wave setting on the Mouse popup), the changes you make will have sharp edges and other high-frequency components. As a result, you will create a wave that spreads out quite a bit as it propagates; there will be an initial wavefront that quickly travels to the edges of the screen, but there will be a bunch of high-frequency noise left behind which will hang around for a while (you can see it better if you turn the brightness up). This would not happen in a nondispersive medium or in a medium where high-frequency waves are faster than low-frequency waves.

The Source popup controls the wave sources. It has the following settings:

• No Sources: there will be no source of new wave motion except for waves you create with the mouse.
• 1 Src, 1 Freq: there will be a single source of sinusoidal waves at a single frequency (set using the Source Frequency slider). This source can be dragged anywhere on the screen with the mouse.
• 1 Src, 2 Freq: the source will be emitting two waves, at separate frequencies. The first frequency is set using the Source Frequency slider, and the second frequency is set using 2nd Frequency.
• 2 Src, 1 Freq: two sources will be created, both at the same frequency. But you can select the phase difference using the Phase Difference slider. If the slider is all the way to the left, the sources will be in phase; if it is all the way to the right, the sources will be 180 degrees out of phase. (The top one will be positive while the bottom one is negative, and vice versa.)
• 2 Src, 2 Freq: the two sources will be at different frequencies. The Source 2 Frequency slider can be used to set the second one's frequency.
• 3 Src, 1 Freq or 4 Src, 1 Freq: 3 or 4 sources will be created, all at the same frequency.
• 1 Src, 1 Freq (Square): the source will emit a square wave. This works best at low freqencies; at high frequencies it is hard to tell it from a sine wave.
• 1 Src, 1 Freq (Pulse): the source will emit positive pulses periodically.
• 1 Moving Src: the source will move, thereby demonstrating the Doppler effect. The speed can be controlled with the Source Speed slider.
• x Plane Src, y Freq: the source(s) will emit plane waves rather than circular waves. The location and direction of the plane wave can be modified by dragging one or both of the two blue circles. If the blue circles are located at the edge of the screen, the plane is extended offscreen; otherwise it is not. If it is not extended offscreen it is finite and so is not a true plane wave, strictly speaking.
• 1 Plane 1 Freq (Pulse): the source will emit plane wave pulses.

The Mouse popup controls what happens when the mouse is clicked. If the popup is set to Mouse = Edit Wave, then a positive or negative area is drawn on the screen. When the mouse is released, this will create a circular wave centered at that point.

If the popup is set to Mouse = Edit Wall, then clicking on a point will create a wall there which will reflect waves. Clicking on a wall will erase it.

If the popup is set to Mouse = Edit Medium, then clicking on a point will create (or remove) an area which has a higher refractive index than the surrounding area, so that waves will move slower through it. This area will be shown in blue.

If the popup is set to Mouse = Hold Wave, then if you click on a point and hold the mouse down, it will create a positive area on the screen which will persist as long as the mouse is down. This will cause the surrounding area to also be positive. For sound waves, this is like adding air at that point; it puts more pressure on the surrounding area.

The Clear Waves button clears out any waves but does not remove any walls or sources. The Clear Walls button clears out walls without clearing out waves.

The Add Border button add walls all around the edge of the screen, so the waves will be reflected at the edges of the screen. If you don't put walls up, then the waves will just drift off the edge of the screen. These walls can be removed with the Clear Walls button; or you can remove some of them with the mouse, if you set the mouse popup to "Mouse = Edit Walls".

The Import/Export button will let you save and load your own creations. Java security restrictions prevent the applet from saving a file directly on your computer. Instead, to save a setup, click this button, copy the selected text, and paste it into another file. To load it again later, click Import/Export again and paste the saved information into the window, and then click Import.

The Stopped checkbox stops the applet, in case you want to take a closer look at something, or if you want to work on something with the mouse without worrying about it changing out from under you.

The Fixed Edges checkbox determines what happens when the wave hits a wall. To simulate sound waves, this should not be checked. To simulate electromagnetic waves or waves in a membrane, this should be checked. If this box is unchecked, waves will be reflected with no phase change (so, positive wavefronts will still be positive when reflected). If it is checked, waves will be reflected negatively (positive wavefronts will be negative when reflected). Different types of waves have different boundary conditions when they hit an obstacle, and that's what determines the behavior when a wave is reflected.

A good example to illustrate this is a string. If you have a string under tension, fixed at either end, then waves going in one direction along the string will be reflected negatively when it hits the end of the string, because the two wavefronts (incident and reflected) have to add up to zero at the end of the string. If the end of the string is allowed to move freely up and down, then a wave will be reflected positively when it hits the end of the string, because the wavefronts no longer have to add up to zero at the edge. A similar argument applies to the two-dimensional case.

The 3-D View checkbox gives you a 3-D view. You can rotate the view by dragging the mouse, but you can't modify the waves or walls in this mode. The brightness slider will adjust the height of the waves.

The Simulation Speed slider controls how far the waves move between frames. If you slide this to the left, the applet will go faster but the motion will be choppier.

The Resolution slider allows you to speed up or slow down the applet by adjusting the resolution; a higher resolution is slower but looks better.

The Brightness slider controls the brightness, just like on a TV set. This can be used to view faint waves more easily.

Now a list of some of the many types of waves simulated by this applet. (Actually the applet only simulates two basic types of waves, but you can interpret the waves as being many different types.)

• Sound waves are longitudinal waves in air or other fluids. The positive areas are high pressure, and the negative areas are low pressure. To simulate sound waves, the "Fixed Edges" checkbox should be unchecked.

  On my computer, at the default settings, a wave takes about four seconds to travel across the screen. In real life, at typical temperatures, sound waves travel about 340 meters/second or about 1100 feet/sec (760 mph).

• Compression waves are longitudinal waves in solids. These are similar to sound waves. This applet does not properly simulate the dispersive effects that would occur in a real solid. (In this applet, low frequency waves go faster than high frequency waves, whereas in a solid it is the opposite.) The positive areas are high compression, and the negative areas are low compression. The "Fixed Edges" checkbox should be unchecked.

• Water waves are actually far more complicated than the simplified wave model used by this applet. But if we keep the amplitude of the waves small, we can pretend that the waves represent water waves. The positive areas are where the water is high, and the negative areas are where it is low. To simulate water waves, the "Fixed Edges" checkbox should be unchecked, because there are no constraints on what the water level should be at the edge of the water.

• A membrane is a thin elastic substance under tension, like a sheet or a square drum head. The positive areas are where the sheet is higher than the edges, and the negative areas are where the sheet is lower. The edges are at a fixed level, so the "Fixed Edges" checkbox must be checked.

• Electromagnetic waves are radiation produced by electric and magnetic fields. Light, radio waves, and microwaves are all electromagnetic waves. The positive areas are where the electric field is in the positive z direction, and the negative areas are where the electric field is in the negative z direction. The magnetic field is not shown. The "Fixed Edges" checkbox should be checked to properly simulate electromagnetic waves.

  On my computer, at the default settings, a wave takes about four seconds to travel across the screen. In real life, electromagnetic waves travel at about 300 million meters/second or about 186,000 mph.

Click here to go to the applet.