In Chapter 6 you learned that one characteristic of simple harmonic motion is the fact that the motion occurs in a pattern that is repeated over equal intervals of time. If the end of a slinky or a stretched string is moved in simple harmonic motion the disturbance produced at the moving end will be propagated along the slinky or string, resulting in a wave. A wave is a periodic disturbance that propagates energy. In the case of mechanical waves such as waves in a slinky, string, or water it is necessary to have a medium through which the energy is propagated, while for electromagnetic waves no medium is needed. The velocity of waves in a string depends upon the tension of the string and upon the mass of the string per unit length of the string. The greater the tension in the string, the faster is the wave velocity; the greater the mass per unit length of the string the slower is the wave velocity.

The frequency, f, of a wave tells us how many vibrations of the wave occur per unit time while the period, T, tells us how long one vibration lasts. Thus the frequency is merely the reciprocal of the period or f = 1 / T. Normally the period is measured in seconds, so the frequency has units of reciprocal seconds, which is given the name Hertz (Hz). If the amount of wave disturbance is plotted versus distance we can identify a regular pattern in the disturbance as is shown in Figures 15.6 and 15.7 on page 304 in the text. The shortest distance between two adjacent positions that have the same displacement from equilibrium is the wavelength, l, (the Greek letter lambda).

The wavelength for a longitudinal wave is shown in Figure 15.4 on page 303 in the text, and the wavelength for a transverse wave is shown in Figure 15.6 on page 304 in the text. The fundamental relationship between the velocity of a wave, its frequency, and its wavelength is expressed as: velocity is equal to frequency times wavelength, or v = f l Thus if we know any two of these quantities for a wave, we can determine the third. For a string we can combine the relationship between velocity, tension, and mass per unit length of the string with the relationship between velocity, frequency, and wavelength to determine the frequency as a function of the tension and the mass per unit length of the string. We obtain an equation that expresses what we observe when we play stringed instruments, such as guitars. If we put more tension on the string, the frequency increases, and we get a higher pitch. If a heavier string and a lighter string have the same tension the lighter string will vibrate at a higher frequency.

For a longitudinal wave the disturbance is along the same direction in which the disturbance is propagated while for a transverse wave the disturbance is perpendicular or transverse to the direction of propagation. Sound waves are longitudinal waves while waves in a string are transverse waves.

Waves involve a moving disturbance that propagates energy. For sound waves the disturbance is the difference in pressure that is produced by something that is vibrating. Molecules are caused to move closer together (or to condense) thereby producing a higher pressure region or a condensation or to move farther apart producing a rarefaction. Our ears are very sensitive to these variations in pressure and identify them as sound. These differences in pressure occur along the same direction in which the sound wave is propagated, so sound is a longitudinal wave. A representation of the condensation and rarefaction of sound waves is shown in Figure 15.16 on page 310 in the text. A plot of the variation of pressure with position is also shown in Figure 15.16.

When two or more waves travel in the same region at the same time, they can interfere with each other. The resulting disturbance is merely the sum of the individual disturbances. If the resulting disturbance at a given location produces a larger disturbance than either of the individual waves produces, we consider the interference to be constructive (a larger disturbance is constructed). If the resulting disturbance is smaller than that of either individual wave we consider the interference to be destructive (the waves tend to destroy each other). This interference of waves can be produced by waves that are sent down a string and by the wave reflected from the other end of the string in such a way that it appears that the wave is "standing" in one place and is not propagating. The position of maximum disturbance in such a condition is called an antinode, and the position where there is no disturbance is called a node. Nodes are one half wavelength apart. Musical instruments are designed to produce standing waves in order to produce louder sounds.

In order to produce a sound that gives the physiological sensation of being louder we must increase the amplitude of the vibration. To produce a higher pitched sound we must have the disturbance occur at a higher frequency. The musical quality of a tone is determined by the number and relative amount of the harmonics of the sound which are present. Thus loudness is associated with amplitude, pitch with frequency, and quality with the number of harmonics present.