Page 195 - Physics Coursebook 2015 (A level)

Chapter 13: Waves
The relationship also implies that, for a particular wave:
intensity2 =constant amplitude
So, if one wave has twice the amplitude of another, it has four times the intensity. This means that it is carrying energy at four times the rate.
QUESTION
5 Waves from a source have an amplitude of 5.0 cm and an intensity of 400 W m−2.
a The amplitude of the waves is increased to 10.0 cm. What is their intensity now?
b The intensity of the waves is decreased to 100 W m−2. What is their amplitude?
Wave speed
The speed with which energy is transmitted by a wave is known as the wave speed v. This is measured in m s−1. The wave speed for sound in air at a pressure of 105 Pa and a temperature of 0 °C is about 340 m s−1, while for light in a vacuum it is almost 300 000 000 m s−1.
The wave equation
An important equation connecting the speed v of a wave with its frequency f and wavelength λ can be determined as follows. We can find the speed of the wave using:
speed = distance time
But a wave will travel a distance of one whole wavelength in a time equal to one period T. So:
wave speed = wavelength period
or
λ v=T
v = ( T1 ) × λ However, f = T1 and so:
A numerical example may help to make this clear. Imagine a wave of frequency 5 Hz and wavelength 3 m going past you. In 1 s, five complete wave cycles, each of length 3 m,
go past. So the total length of the waves going past in 1 s is 15 m. The distance covered by the wave in one second is its speed, therefore the speed of the wave is 15 m s−1.
Clearly, for a given speed of wave, the greater the wavelength, the smaller the frequency and vice versa. The speed of sound in air is constant (for a given temperature and pressure). The wavelength of sound can be made smaller by increasing the frequency of the source of sound.
Table 13.1 gives typical values of speed (v), frequency (f ) and wavelength (λ) for some mechanical waves. You can check for yourself that v = f λ is satisfied.
Water waves in a ripple tank
Sound waves in air
Waves on a slinky spring
about 0.12
about 6
about 0.2
about 300
20 to 20 000 (limits of human hearing)
15 to 0.015
about 1
about 2
about 0.5
Table 13.1 Speed (v), frequency (f) and wavelength (λ) data for some mechanical waves readily investigated in the laboratory.
WORKED EXAMPLE
2
Middle C on a piano tuned to concert pitch should have a frequency of 264 Hz (Figure 13.10). If the speed of sound is 330 m s−1, calculate the wavelength of the sound produced when this key is played.
Step1 Weusetheaboveequationinslightly rewritten form:
wavelength = speed frequency
Step2 SubstitutingthevaluesformiddleCweget: wavelength = 330 = 1.25 m
The human ear can detect sounds of frequencies between 20 Hz and 20 kHz, i.e. with wavelengths between 15 m and 15 mm.
wave speed = frequency × wavelength v=f×λ
Speed / m s−1
Frequency / Hz
Wavelength /m
264
Figure 13.10 Each string in a piano produces a different note.
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