There are many types of waves: light waves, sound waves, radio waves, cosmic rays, x-rays, communication waves, waves on cords, etc. In our first discussion of waves, we will deal with that type which is called just
"wave", that is, a water wave.
Let us assume that a stone is thrown into the middle of a large, calm pond on a day when there is no wind. If there is a perpendicular plane surface cutting the water surface through a point where the stone hits the water, an observer would see the water surface disturbed in such a way that a curve would be visible. This curve would have a shape as shown in figure 5.2. In figure 5.2 it is important to note that the pattern of crests
and troughs is moving. If the stone hits the water Figure 5.2: Waveform dimensions
surface at the point (P), the pattern is moving to the right in the diagram above. Of course, the entire pattern is moving out from point (P) in all directions, but we are looking in only one direction. We should also note that the pattern is moving with a definite speed, called the wave speed (v).
The amplitude (A) of the wave is the greatest displacement from the rest position. The amplitude is shown in figure 5.2.
Another distance that we will need in our discussion of waves is the wavelength, λ. (Greek letter lambda).
The wavelength is defined as the distance from one point on the wave pattern to the next point in a similar position. The distance from the top of a crest to the top of the next crest is a wavelength. Also the distance from the bottom of one trough to the bottom of the next trough is also the same distance, one wavelength.
The distance λ, is also shown in the diagram.
Let us next consider sinusoidal wave motion impressed on a very long flexible cord by an oscillating body.
Assume that the oscillating body is a sphere attached to a vertical spring.
After the spring has been oscillating for some time, the physical situation is as shown in figure 5.3.
The frequency (f) of the oscillating body is defined as the number of complete oscillations in one second.
Frequency is expressed in cycles/sec, or Hertz. The period (T) is defined as the time for one complete oscillation. It is expressed in seconds.
Let us suppose that the oscillating body completes 6 oscillations in one second. It follows that the time for one oscillation is one-sixth of a second.
In this case:
F=6Hz and T=
Figure 5.3: A waveform produced on a piece of string by a mass oscillating on the end of a spring From this example we see that f and T are reciprocals of each other.
T = and f =
We next seek a relationship between wave speed (v), frequency (f), and wavelength (λ,).
We note that the wave moves forward a distance of one wavelength in a time of one period. Of course, the wave moves with speed (v).
Since the distance equals the speed times the time, we can write the equation:
λ=νΤ From this equation, we have:
ν And finally:
fλ = ν
EXAMPLE:
(a) A body oscillates with a frequency of 8 Hz, and sends out a wave having a wavelength of 0.2 ft. What is the speed of the wave?
V = (8 cycles/sec.) (0.2 ft.) = 1.6 ft./sec.
(b) What is the wavelength of a wave moving with a speed of 5 ft./sec. If the frequency of the oscillating body which is the source of the wave is 12 Hz?
λ = = = 0.417 ft
(c) An observer times the speed of a water wave to be 2 ft./sec. and notes that the wavelength is 0.5 ft. What is the frequency of the disturbance that gives rise to this wave?
f = = 4 cycles/sec. = 4 Hz
ν=f λ
Resonance
In the case of water waves and in the case of waves on a very long cord, we were able to neglect waves long cord a sinusoidal wave travels continuously along the cord. However, if the sinusoidal wave meets a fixed end, a reflected wave moves back along the cord.
The wave patterns which are observed are called the normal modes of vibration of the cord. In figure 5.4.
the length of the cord is L. The wavelength in the various modes of vibration are X. The n is the index of the mode. In the equations which follow, n has an integral value, that is n = 1 , 2, 3, 4.
We can write a general relation as follows:
λn =
The vibration where n = 1 is called the fundamental mode of vibration of the body. The other vibrations are called overtone vibrations. Every body which can vibrate has a certain fundamental mode of vibration of a definite frequency. If this frequency is impressed on the body, it will vibrate with a relatively large amplitude.
We say that the body is vibrating in resonance with the impressed frequency.
Figure 5.4: Normal modes of vibration Problems
1. A water wave has a wavelength of 0.9 ft. and the wave speed is 4.5ft/sec. What is the frequency of the disturbance setting up this wave?
2. A wave on a cord is set up by a body oscillating at 12 Hz. The wavelength is 0.25ft. What is the wave speed?
3. A water wave is set up by a source oscillating at 12 Hz. The speed of the wave is 24ft/sec. What is the wavelength?
Sound waves are usually defined as pressure waves in air or in some other material medium. Sound waves originate in some vibrating body such as the oscillation of a person's vocal cords or the periodic rotation of a plane's propeller.
As the source of sound vibrates, the air surrounding the source is periodically compressed and rarefied (made less dense). This periodic change in the atmospheric pressure moves forward with a definite speed of propagation called the "speed of sound".
The speed of sound in air is dependent on the temperature of the air. This is not surprising since the molecules of air move faster in their random motion if the temperature is higher. Thus we should expect these pressure waves to move somewhat more rapidly in warmer air.
The speed of sound in air is approximately 331.5 m/s at 0°C
At an air temperature of 20°C, the speed of sound increases to 344 m/s
If an ear and its eardrum are in the vicinity of a sound wave, the air which strikes that eardrum has a periodically changing atmospheric pressure. If the frequency of the sound is middle C (256 Hz), and the atmospheric pressure that day is 14.7 Ibs/in2, 256 times each second the air pressure is slightly above 14.7 Ibs/in2 and 256 times each second the pressure is slightly below 14.7 Ibs/in2 it should be emphasized that
"slightly" means very small. The human ear is a remarkably sensitive instrument. It can detect air pressure variations as small as about 0,000000005 lbs./in.2
Sound travels faster in liquids, and even faster still, in solids.