Sound waves differ from waves in a string in a fundamental way. While the string wave travels horizontally along the string, the individual particles of the string move vertically at right angles to the direction of wave travel. This is not the case with sound. While a sound wave travels horizontally, the motion of the air molecules is also horizontal.

string waves:	sound waves:

Here are two names you should know. The wave in which particle motion is perpendicular to wave direction is known as a transverse wave. The wave in which particle motion is parallel to wave direction is known as a longitudinal wave. Thus waves on water and waves in a flag are transverse, while sound waves and pulsing waves down a spring are longitudinal.

We may hold down the end of a string to prevent it from vibrating. Likewise, we may prevent motion of a sound wave by having it strike a solid wall. This will effectively prevent horizontal motion of the air molecules. A sound wave vibrating between two walls, then, is analogous to a string vibrating between two solid supports. We may draw the modes of oscillation similarly for either case:

String:
Sound:

Remember, a longitudinal wave is difficult to draw, so we sketch it as if it were transverse.

If we open up one end of a tube in which sound waves are vibrating, we get free motion of the sound wave, and it is analogous to the bathtub. Here are the harmonics for a tube open at one end, closed at the other.

First mode	Second mode	Third mode

Here are the harmonics (modes) for a pipe open at both ends.

First mode	Second mode	Third mode

8. Organ pipes are open at the end where the tongue is. The other end may be open or closed. Find the wavelength of the third mode of vibration for a closed organ pipe 3.7 m long.

1.25£3 = 3.7 m £3 = 2.96 m

9. Find the wavelength of the 2nd mode of oscillation for an open organ pipe 2.6 m long.

£2 = 2.6 m

Of course when we listen to an organ or a guitar, we are not too concerned with the wavelength of sound produced. We are more concerned with the frequency, because that is what we hear.

Here's the relationship between frequency and wavelength: imagine a wave vibrating at 10 cycles/second, sending out a wave 2 meters long. After one second it will have sent out 10 waves, each 2 meters long, or a 20 meter long train of waves. Thus the first wave has traveled 20 meters in 1 second. In general, we may say

v = f£

This makes sense. Imagine a wave being drawn on a paper by a moving hand vibrating slowly and then by a moving hand vibrating quickly.

Slowly vibrating sine wave	Quickly vibrating sine wave

The high frequency results in a short wavelength. The two are inversely related as the equation, v = f£, indicates.

10. An ocean wave travels at 6.0 m/sec, and has a frequency of 0.20 cycles/sec (hertz). What is its wavelength?

v = f£ £ = v/f £ = (6 m/sec)/(.2 cycles/sec) £ = 30 m

11. A wave in a string travels at 18 m/sec and has a wavelength of 0.80 m. What is its frequency?

v = f£ f = v/£ f = (18 m/sec)/(.8 m) f = 22.5 cycles/sec

12. A bathtub transmits waves at 1.4 m/sec and is 1.9 m long. What is the frequency of its first mode?

£/2 = 1.9 m £ = 3.8 m f = v/£ f = (1.4 m/sec)/(3.8 m) f = .368 cycles/sec

13. A guitar string is 1.2 m long. If its 3rd harmonic is 840 hz, what is the speed of a wave along the guitar string?

(3/2)£3 = 1.2 m £3 = .80 m v = f£ v = (840 cycles/sec)(.8 m) v = 672 m/sec

14. A guitar string is 2.3 m long and waves travel along it at 97 m/sec. What is the frequency of its first mode of vibration?

£/2 = 2.3 m £ = 4.6 m f = v/£ f = (97 m/sec)/(4.6 m) f = 21.1 cycles/sec

A note on units The terms ";cycles/second"; or ";vibrations/second"; are for convenience. In fact, cycles and vibrations have no dimensions they are merely the result of counting. Thus the units for frequency should be 1/sec or sec-1. The name I've used for this unit, the hertz, is named after Heinrich Hertz, the guy who discovered radio waves. One hertz (hz) is a cycle/second, a kilohertz (khz) is 103 cycles/sec, and a megahertz (mhz) is 106 cycles/sec.

15. A closed organ pipe (remember, that means open at one end, closed at the other) 0.82 m long produces a 317 hz tone in its second mode of vibration. What is the speed of sound in the pipe?

(3/4)£2 = .82 m £2 = 1.09 m v = £2f2 v = (1.09)(317) v = 347 m/sec

The speed determined in the previous problem is a little higher than normal. Although sound speed is dependent on pressure, temperature and density, we will take 330 m/sec as a good, typical speed of sound.

16. Find the frequency of the 3rd mode of vibration in an open organ pie 3.7 m long.

3.7 m = (1.5)£3 £3 = 2.47 m f3 = v/£3 f3 = (330 m/sec)/(2.47 m) f3 = 134 hz

17. If the frequency of the 2nd mode in a closed organ pipe is 514 hz, what is the frequency of the first mode?

£2 = v/f2 £2 = (330)/(514) £2 = .642 m £1 = 3£2 £1 = 1.93 m f1 = v/£1 f1 = (330)/(1.93) f1 = 171 hz

18. Dwain D. Bafftub finds that his shower supports a first mode vibration of 57 hz when he sings in it. His shower is 2.7 m high. What is the speed of sound in his shower? (It is somewhat different from normal because of differences in temperature and humidity.)

£/2 = 2.7 m £ = 5.4 m v = f£ v = (57)(5.4) v = 308 m/sec note: the shower represents a closed organ because the floor and ceiling prevent vibration at the ends.

Cultural note: It has been found that men, far more than women, like to sing in showers. Presumably this is because showers are of the right size to reinforce low notes, supporting the latent macho feelings of the singers.

19. The third mode of a closed organ pipe sounds at 550 hz. What is the length of the pipe?

(1.25)£3 = L £3 = v/f3 £3 = (330 m/sec)/(550 hz) £3 = 0.60 m L = (1.25)£3 L = (1.25)(0.60 m) L = 0.75 m

20. A guitar string, 0.84 m long, sounds its 2nd mode at 110 hz. What is the velocity of the wave in the string?

£2 = L £2 = .84 m v = f£ v = (1100)(.84) v = 924 m/sec

Before we go on, you may have wondered how you can observe different modes of vibration in your own home. You can get a jug or a pip bottle and blow over the top to excite the first mode for a pipe open at one end and closed at the other. Higher modes can be heard by blowing hard, directing the air stream at the bottle rim opposite the one against your lips. It takes a little practice.

You may more easily observe the same thing by blowing harder into a recorder or other simple wind instrument. The tone will abruptly jump to a higher frequency.

A guitar may be sounded at the second mode by lightly touching the string right in he middle and plucking it gently with the other hand. The technique can also be used in tuning the instrument.

21. The second harmonic in an open organ pipe is 2,400 hz. What would be the second harmonic in the same pipe if one end were closed?

L = £2 L = v/f2 L = (330)/(2400) L = .138 m (3/4)£2' = .138 m £2' = .183 m f2' = v/£2' f2' = (330)/(.183) f2' = 1800 hz

22. What is the fundamental frequency of a 1.6 m long bathtub if the wave velocity in it is 3.0 m/sec? ...what is the 3rd harmonic?

£1/2 = 1.6 m £1 = 3.2 m f1 = v/£1 f1 = (3)/(3.2) f1 = .938 hz (3/2)£3 = 1.6 m £3 = 1.07 m f3 = v/£3 f3 = (3)/(1.07) f3 = 2.80 hz