Chapter 19^*: Vibrations and Waves

Waves are a form of energy in that they have the ability to perform work. They are also messengers as they carry information that disturbances in the Universe have occurred. For example, sound waves are caused by pressure disturbances, electromagnetic waves are caused by oscillating charges, water waves are caused by wind or disturbances in or on the water, gravitational waves result from the motion of mass, and so on. All of these waves may induce motions in objects when they interact.

To understand waves, let us set some jargon:

• vibrations are periodic oscilations in time performed by an object
• waves are periodic variations that occur in space and time--waves are spread out in space and time, they are not localized objects such as are particles.

In this lecture we describe:

• waves properties such as wavelength, frequency, and wave speed
• transverse waves versus longitudinal waves
• the wave phenomena interference, standing waves, and the Doppler Effect
• Shock Waves, Bow Shocks and Bow Waves, and Tsunami

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^* Chapter 20: Sound utilizes many of the concepts described here in the context of sound waves. The two discussions blend. These comments apply also to Chapter 21: Musical Sounds.

Vibrations, Waves, Clocks, and the Meter Stick

Pendula and Vibrations: To the right is shown a pendulum. The period of swing for a pendulum depends only on its length (for small swings)!!! The period of swing is P = 2 (L/1 meter)^1/2 seconds

where P stands for the period of swing and L is the length of the pendulum. So, for a pendulum of length L = 1 meter, the swing period is ~ 2 seconds. Question: If a pendulum has a swing period of 1 seconds, how long is the arm of the pendulum? What about for a swing period of 20 seconds? This periodicitiy has been taken advantage of to make clocks. Other natural vibrations that have been used in clocks are the regular vibrations crystals undergo when jostled.

Waves and Clocks: The distance between the crests of wave defines its wavelength while the rate at which the crests fly by an observer defines its frequency. It is clear that there is a relationship between the wavelength W, the frequency f, and the speed at which the wave move v. Formally, we have v = W x f.

The electromagnetic radiation produced by Cesium atoms today make exceedingly accurate clocks, Atomic Clocks and the speed at which electromagnetic radiation travels in vacuum (300,000 kilometers per second) is used to define our unit of length measurement, the meter.

Compression (Longitudinal) Waves and Transverse Waves

Waves may be classified as longitudinal waves and transverse waves. The distinction can be understood as follows. Suppose a wave propagates to the right (see top right). In both cases, assume that the wave moves from left-to-right. If the wave moves as a series of compressions and expansions (rarefactions) where the expansion and compression are in the same direction the wave moves ===> Longitudinal (Compression) Wave. If the rope is disturbed up and down, say like a pulse on the rope ===> the disturbance is perpendicular to the direction in which the wave moves ===> Transverse Wave. Sound is a longitudinal (compression) wave and electromagnetic radiation is a transvese wave. Both sound waves and electromagnetic radiation travel at speeds that are indepedent of the wave frequency. Some wave speeds for various phenomena.

• Sound: v = (1.4 P / density)^1/2 ~ 350 m/s for atmospheric air at 300 Kelvin (27 Celsius)

• EM radiation: 300,000 km per second

• waves on a rope: v = (T / l)^1/2, where T is the tension is the mass per unit length of the rope

• Tsunami: v = (g x depth)^1/2, where g is the gravitational acceleration = 9.8 m per s². For a depth of 1 kilometer, v ~ 360 km/hr

A nice application of wave propagation and its use is the seismic mapping of the interior of the Earth where both transverse and longitudinal waves are generated.

Interference, Beats, Standing Waves, and Doppler Effect

Interference

The Principle of Superposition applies to many waves we encounter in our everyday lives. This makes understanding how colliding waves behave, straightforward and fairly easy. When two low-amplitude waves that obey the Principle of Superpostion meet, the waves simply add. The new wave takes on double the amplitude if the crests of the meeting waves coincide (constructive interference if they are in-phase) or they subtract if one's peak and the other's trough coincide (destructive interference if they are 180 degrees out-of-phase). If the waves have the same amplitudes and frequencies, then

Beats

When two waves interact that have slightly different frequencies, an effect known as beating arises. The two waves interfere but there is a lower frequency amplitude variation that happens. The peaks and troughs do not line up each cycle, rather, they line-up only after a number of cycles has passed.

The waves line-up with frequency

f_beat = f₁ - f₂

For two waves with nearly the same frequency, the beat frequency is very low. For f₁ = 440 cycles per second and f₂ = 438 cycles per second, the overall amplitude (envelope) modulation, the beat occurs with frequency 440 cps - 438 cps = 2 cps where, for convenience, we abbreviated cycles per second with cps. Generally, one cps is referred to as 1 Hertz, abbreviated as Hz.

Examples

Two wave sources separated by 1 wavelength, 2 wavelengths, and 4 wavelengths (top three pictures). The bottom is for two wave sources separated by the same amount as the 2nd plot, but the wave frequencies are in the ratio 5:4.

Standing Waves

Two waves moving in opposite directions interfere and set up a standing wave. The wave oscillates but it does not travel. We see that the points where the wave are zero ( Nodes) are stationary. How do Standing Waves Arise?

At the wall on the right, the wave's amplitude is forced to zero which causes the wave to flip upon reflection. At the wall on the right, the wave's amplitude is allowed to float which causes the wave to bounce with the same sense of amplitude upon reflection. The wave then interferes with itself to set-up the Standing Wave.

Different Modes of Oscillation (different wave frequencies). Again notice that the nodes are stationary. The crests peaks of the combined waves are Antinodes.

Modes of Oscillation (different wave frequencies) on a drumhead. There are now stationary lines where the wave amplitude is zero .

Doppler Effect

If the source of waves is in relative motion with respect to the observer, the wavelengths for the waves measured by the observer will be shifted. If the source of the waves is approaching, the wavelength will shortened (Blueshift) if the source is moving away from the observer the observed wavelength will be lengthened (Redshift). The size of the shift depends on the speed of the source of the waves, V, and the speed at which the wave travels, v.We can easily show that

(W_obs-W_true) = V/v x W_true

This the classical Doppler Effect.

The wave generator launches a series of equi-spaced concnetric circles. The distance between the crests is constant if the waves are generated at regular intervals. No wave interferes with another.	The wave generator moves to the right and so launches a series of circles whose centers are equi-spaced in the direction in which the wave generator moves. The distance between the crests of the waves varies depending upon whether you are in front of the source or behind the source. This is the Classical Doppler Shift. As long as the soure of the waves moves more slowly than the waves, no wave still interferes with another.	An analogous effect occurs for electromagnetic radiation. A directed beam of microwaves with wavelength Wo aimed at an approaching car, reflects off the approaching car. The reflected beam of microwaves returns to the observer with a shortened wavelength Wr . The amount of compression depends on the speed of the moving car (see the middle panel above). Recall that the relative change in the wavelength is given by Relative Change = (Wr- Wo )/Wo) = (source speed)/wave speed) Microwaves, electromagnetic radiation travel at the speed of light, c = 300,000 km/s = 1,080,000,000 km/h! A car moving at 100 km/h would then lead to a relative shift in the wavelengths of the returning microwaves of 0.000000093. A remarkably small shift, but detectable. This is why we don't see Doppler shifted colors in our everyday lives.