13 WAVE MOTION 13.6 The Doppler effect
car. These waves reflect off the car, which effectively becomes a moving source.
Hence, by measuring the frequency increase of the reflected waves, the policeman
can determine the car’s speed.
Worked example 13.1: Piano range
Question: A piano emits sound waves whose frequencies range from fl = 28 Hz to
fh = 4200 Hz. What range of wavelengths is spanned by these waves? The speed
of sound in air is v = 343 m/s.
Answer: The relationship between a wave’s frequency, f, wavelength, λ, and
speed, v, is
v = f λ.Hence, λ = v/f. The shortest wavelength (which corresponds to the highest
frequency) is
v
λl =
fh
343
=
4200= 8.1667 × 10 −^2 m.The longest wavelength (which corresponds to the lowest frequency) is
v
λh =
l343
=
28= 12.250 m.Worked example 13.2: Middle C
Question: A steel wire in a piano has a length of L = 0.9 m and a mass of m =
5.4 g. To what tension T must this wire be stretched so that its fundamental
vibration corresponds to middle C: i.e., the vibration possess a frequency f =
261.6 Hz.
Answer: The fundamental standing wave on a stretched wire is such that the
length L of the wire corresponds to half the wavelength λ of the wave. Hence,
λ = 2 L = 1.80 m.f