We are discussing waves in general. We have defined terms that describe properties of waves in general.
We will continue this chapter and discuss some additional general properties of waves.
The speed of a wave traveling along a string is determined by the tension, F, in the string and the mass per unit length, m, of the string, v = Sqrt{F/m}. Dimensional analysis shows that the units of the right hand side are length/time, the same as velocity.
A wave traveling in the positive x direction has a frequency of 25.0 Hz, as in Figure P13.32. Find the (a) amplitude, (b) wavelength, (c) period, and (d) speed of the wave.
(a) The amplitude is half the crest to trough ( or peak to peak) distance of 18cm, or A = 9.0cm.
(b) The 10 cm distance shown in the figure is one half a complete cycle, from a crest to a trough, thus wavelength is twice this distance, or l = 20 cm
(c) Since f=25.0 Hz, T = 1/f = (1/25)s = 0.0400 s.
(d) Use v = fl = (25.0 Hz)(0.20 m) = 5.0 m/s.
Waves obey the super-position principle:
The waves used in this example have the following characteristics:
If we imagine these two waves a short time later when the crests of one line up with the troughs of the other, then they will like like: [diagram]. The resultant wave is again found by adding the displacements point by point. This time the sum cancels at every point -- the displacement of the first wave at any point is exactly opposite the displacement of the second wave at the same point! We say that these waves are "out of phase", and that they interfere destructively.
The super-position principle can be used for any combination of waves. For instance, Figures 13.31 and 13.32 show how 2 pulses of different shape can constructively or destructively interfere.
And the super-position principle works in the reverse sense: a wave of arbitrary shape can be imagined as being formed from the sum of many "simple" waves. This fact will be seen more as we discuss sound waves.
What happens to a wave when its medium ends? For instance, when a pulse on a rope reaches the end of the rope, what happens to it? Or if I speak so that my voice is directed at a wall, what happens to the sound waves when they reach the wall?
In both cases, the waves reflect. Exactly how they reflect depends on how the medium ends. The rope can be tied tightly to a post, or it can be secured to a ring so that it is free to slide up and down the post.
When it is tied tightly, then that end of the rope cannot move. Consequently, the reflected pulse will be the negative of the incoming pulse such that the two cancel at the fixed end. [see Fig. 13.33]
When it is fixed so that it is free to slide, then the reflected pulse will not be inverted. The sum of the incoming and reflected pulse causes the free end to displace by double the displacement of the pulse. [See Fig. 13.34]
Now we will move from a general discussion of waves to discuss some phenomena specific to sound waves.
Sound waves are longitudinal waves traveling through air (or sometimes other media) as variations in density or pressure. The pattern of alternating high and low density carries the sound that we hear. Regions of high density or pressure are called compressions, while regions of low density or pressure are called rarefactions. In air, sound is commonly produced by a vibrating object which alternately compresses the air, then rarefies the air. And these compressions and rarefactions result in a sound wave. Examples of vibrating objects are the tines of a tuning fork, your vocal cords, and the cone of a speaker.
Sound waves are longitudinal waves, meaning that the displacement of air molecules is parallel to the direction of propagation. Recall that waves where the displacement is perpendicular to the direction of propagation are called transverse waves.
We can normally hear sound in the frequency range of 20Hz to 20,000Hz. We call these audible waves. Sound waves at frequencies below 20Hz are called infrasonic, and those above 20,000Hz are called ultrasonic.
The speed of sound in a liquid or gas is given by:
v = Sqrt{B/r}
where r is the density of the fluid and B is the bulk modulus (a topic we skipped in Chapter 9).
Sound can also propagate through a solid.
The speed of sound in a solid is given by:
v = Sqrt{Y/r}
where Y is the Young's modulus for the solid (again, refer to Chapter 9).
The speed of sound in air is 331m/s at 0°C.
The density of air changes with temperature, and to correct for this effect we will use the following approximate relation:
vair = (331 m/s)Sqrt{1 + (Tc/273)}
where Tc is the temperature in Celsius.
Sound intensity is measured in decibels, a name derived from Alexander Graham Bell.
The decibel level, b, is given by:
b = 10 log{I/I0}
where log is the base 10 logarithm, I is the intensity of the sound in W/m², and I0 = 1.0x10-12 W/m² is the intensity threshold for hearing.
Table 14.2 lists the decibel levels for a range of sounds.