Lecture 32
• Frequency and Wavelength
• Transverse and Longitudinal Waves
• Speed of Sound
Cutnell+Johnson: 16.1-16.8
Frequency and Wavelength
All kinds of waves have two characteristics. They are traveling disturbances, and they carry
energy. A water wave is disturbance on the surface of the water – some parts are higher than
others. The location of the high parts moves around. This is what I mean by a traveling
disturbance. Obviously a water wave carries energy, because the higher parts have kinetic
energy, which is moving around.
In this class, we’ll mainly focus on periodic waves. These are waves where the disturbance
repeats itself in time. The period of the wave T is the time it takes to repeat itself. The frequency
of a wave is denoted by f , and is
1
f=
T
This idea is the same as simple harmonic motion, or motion in a circle, where the motion also
repeats itself. However, for waves we generally use a special unit for frequency, called the Hertz
(Hz). It is defined by
cycle
1 Hz = 1
sec
In other words, if ten peaks of the wave pass by in a second, the frequency is 10 Hz.
With a periodic wave, you can also define the wavelength λ. This is the distance between
successive peaks (or successive troughs, or successive zeroes – it doesn’t matter which characteristic you pick since the wave repeats itself exactly each time). The wavelength is a distance, so
the MKS unit is of course meters.
There is a simple relation between the frequency, the wavelength, and the speed of the wave.
It is simple to derive. If the distance between the peaks is 5 cm, and 10 peaks go by every second,
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then the wave is traveling at 50 cm/s. The general formula is
v = fλ
For example, this now lets us get the wavelength of radio waves. Radio waves are a form of
electromagnetic energy, like visible light. Their speed is 3.0 × 108 m/s. The frequency of an FM
radio wave is about 100M Hz = 100 × 106 Hz = 108 Hz (FM 91.1 means 93.1 M Hz). This means
that the wavelength is about 3 m long. This is why FM antennas tend to be long and stretched
out – they’re trying to pick up as much of the wave at any given moment as possible. On the
other hand, your AM dial measures frequencies in KHz, AM 1000 means 1000KHz = 106 Hz.
Thus an AM wavelength is about 100 times larger than that of an FM wavelength.
Transverse and Longitudinal Waves
One simple question you can ask of a wave is: what is waving? For a guitar, obviously the
string is moving up and down. For light, the question is fairly subtle, and a real answer requires
understanding Einstein’s theory of relativity.
But for sound, the answer is simple. It is the matter like air, or metal, or whatever, in
between the sound emitter and the sound receiver which is waving. This eventually makes the
air in your ear wave back and forth, which makes your ear drum move back and forth. Your
brain then converts this to sound. You must have some material to convey the sound – in space
no one can hear you scream.
Now that we understand what is waving, we can understand how it waving. There are two
categories of waves: transverse waves and longitudinal waves. A given wave can be one or the
other, or both. A transverse wave is one where the disturbance is perpendicular to the direction
of the wave. A familiar example is a guitar string. The wave is traveling back and forth along
the guitar, but the guitar string is just vibrating up and down. The same is true if you wiggle a
rope tied to a wall. The wave looks like it is traveling back and forth, but the rope is really just
going up and down. One important kind of transverse wave is light.
In longitudinal waves, the disturbances are in the same direction as the direction of the wave.
Like a transverse wave, they still move back and forth, however. Sound is a longitudinal wave.
The speed of sound
The speed of the wave depends only on properties of the thing which is waving: the guitar
string, or the air in the case of sound. It does not depend on the amplitude of the wave, i.e. how
hard you pluck the string, or how loudly you shout.
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I’m sure everyone has noticed the effects of the speed of sound. An echo is one of the effects
– the sound takes a non-zero amount of time to bounce off a surface and come back. An echo
chamber has surfaces which bounce the sound back without too much loss. Thus even though
an echo chamber is small, the sound can bounce back and forth many times before fading away.
One use of the speed of sound is to measure how far away lightning is. The speed of sound
is quite large – 3.0 × 108 m/s. Light can go around the earth multiple times in a second. Thus
the time it takes for lightning to reach you is extremely small, and you can take it to be zero.
However, the speed of sound is much smaller. Thus if you know the speed of sound, you can
figure out how far away the lightning is, by measuring the time difference between the lightning
flash and the thunderclap.
So what is the speed of sound? It depends on the material conveying the sound. First, let’s
study sound moving through a gas like air. The speed of sound is approximately the speed of
the molecules themselves. The reason is that the only way for sound to be carried is for one
molecule to hit the next molecule, and so on. Obviously this can’t go faster than the molecules
are moving themselves. We actually already know the average speed of molecules in an ideal
gas. This comes from our expression for the average kinetic energy...
Thus the speed of sound in a gas is
vsound ∝
r
kT
m
where m is the mass of the individual molecules and T must be measured in Kelvin. The constant
of proportionality is much more difficult to derive. It turns out to be related to the specific heat
of the gas (or actually, the specific heats: it turns out that for a gas the specific heat at constant
pressure is not the same as the specific heat at a constant volume). Anyway, like most of the
constants in this class, you can look it up. The upshot is that the speed of sound in air at 0o C is
331 m/s, while it is 343 m/s at room temperature. It’s easy to check these numbers that these
numbers are consistent with the formula, because 0o C = 273K, while 20o C = 293K. Thus
r
293
343
v293
=
= 1.036 =
v273
273
331
Sound is about 3 times faster in helium, because helium is much lighter. Thus sound travels
about 800mph. If a thunderclap is one second after the lightning flash, the lightning is only
≈ 350 m away. Even a relatively close distance of a mile away will take nearly 5 seconds for the
sound to get there.
The reason you sound funny after you breathe helium is that the velocity of sound is different
in helium than it is in air (it’s much higher). The wavelength of your vocal cords doesn’t change,
so the frequency of the resulting wave is much higher. Of course, the wave crosses from the
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helium in your voicebox back into the atmosphere. However, when it does so, the frequency
does not change, even though the velocity does. The wave is wiggling back and forth at some
number of cycles per second in the helium, and at the boundary between the helium and the air,
it still must be wiggling at the same number of cycles per second. Thus it wiggles at the same
frequency in air as well, so its wavelength gets longer.
There is also a formula for the speed of sound in a solid. Remember how a bar stretches
when a force is applied:
F
∆L
=Y
A
L0
where Y is Young’s modulus. Thus you would think that the larger the Young’s modulus, the
faster the sound goes. Moreover, like for a gas, you expect that the larger the mass, the slower
the sound (it takes more energy to move the particles). The answer is that the speed of sound
in a solid is
s
Y
vsound =
ρ
where ρ is the density. Plugging in the numbers shows that the speed of sound in a solid is much
faster than this in a gas: for example in steel it is nearly 6000m/s.
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