Difficulty: Intermediate | Prerequisites: Sections 15.1 through 15.4 (wave model, wave speed, displacement equations). Logarithms are needed for the decibel scale.
Sections 15.5 through 15.7 move from describing waves in the abstract to describing how sound behaves in the real world: how its energy spreads with distance (intensity and the inverse-square law), how we measure loudness (the decibel scale), and what happens to the perceived frequency when the source or observer is moving (the Doppler effect). The Doppler effect in particular ties together everything you have learnt about wave speed and frequency, and it is one of the most commonly tested topics in introductory physics. If you are comfortable with the four Doppler scenarios (moving observer approaching/receding, moving source approaching/receding), you are well placed for the exam.
Sound intensity drops with the square of the distance from the source, and we measure it on a logarithmic decibel scale. When a source or observer moves, the perceived frequency shifts: higher when closing, lower when separating (the Doppler effect). Reflected sound from a moving object (echo) also undergoes a frequency shift.
Intensity (I)
The power delivered per unit area: I = P_source / area. For a point source radiating uniformly, I = P_source / (4 pi r squared). In simple terms, it tells you how much sound energy hits each square metre of surface at a given distance.
Decibel (dB)
A logarithmic unit for measuring sound intensity relative to a reference level. beta = (10 dB) log_10(I / I_0), where I_0 is the threshold of hearing. Think of it as a way to compress the enormous range of human hearing into manageable numbers.
Power (P)
The rate of energy transfer: P = E / delta t, measured in watts (W).
Inverse-square law
Intensity drops as 1 / r squared for a point source radiating uniformly in all directions. Double the distance and the intensity drops to one quarter.
Doppler effect
The change in observed frequency when there is relative motion between the source and the observer. The observed frequency increases when they move toward each other and decreases when they move apart.
Echo (reflection of sound from a moving object)
When sound bounces off a moving object, the reflected wave has a shifted frequency. The shift depends on the object's speed and direction relative to the wave source.
A point source (e.g. a loudspeaker, a lightbulb) radiates energy outward in spherical wave fronts.
Very far from the source, small segments of the spherical fronts appear flat, so they can be treated as plane waves.
Intensity is power per unit area: I = P_source / (4 pi r squared).
The decibel scale converts this to a more intuitive number: beta = (10 dB) log_10(I / I_0).
Because it is logarithmic, every increase of 10 dB corresponds to a tenfold increase in intensity.
Observer approaching a stationary source: the observer runs into wave crests more frequently than they would at rest, so the observed frequency goes up.
f_observed = (1 + v_o / v_w) f_source
Observer receding from a stationary source: the observer moves away from oncoming crests, so the observed frequency goes down.
f_observed = (1 - v_o / v_w) f_source
v_o is the observer's speed, v_w is the wave speed in the medium.
Source approaching a stationary observer: the source "chases" its own wave crests, compressing the wavelength ahead of it. The observed frequency increases.
f_observed = f_source / (1 - v_s / v_w)
Source receding from a stationary observer: the wavelength behind the source is stretched. The observed frequency decreases.
f_observed = f_source / (1 + v_s / v_w)
v_s is the source's speed.
When sound reflects off a moving object, the frequency shift is approximately:
delta f = +/- 2 f_source (v_o / v_w)
If the object moves closer, the reflected frequency increases (positive shift).
If the object moves away, the reflected frequency decreases (negative shift).
This principle is used in radar guns, medical ultrasound and sonar.
Sound intensity (point source, uniform radiation):
I = P_source / (4 pi r squared)
Decibel level:
beta = (10 dB) log_10(I / I_0)
where I_0 = 10 to the power of negative 12 W/m squared (threshold of hearing).
Power:
P = E / delta t (energy per unit time, in watts).
Doppler: moving observer, stationary source:
f_observed = (1 + v_o / v_w) f_source (approaching)
f_observed = (1 - v_o / v_w) f_source (receding)
Doppler: moving source, stationary observer:
f_observed = f_source / (1 - v_s / v_w) (source approaching)
f_observed = f_source / (1 + v_s / v_w) (source receding)
Echo frequency shift (reflection from a moving object):
delta f = +/- 2 f_source (v_o / v_w)
Positive when the object moves closer, negative when it moves away.
Police radar guns and speed cameras work by bouncing a microwave signal off a moving vehicle and measuring the Doppler shift of the reflected signal. The echo formula gives the vehicle's speed directly.
Medical Doppler ultrasound uses the same principle to measure blood flow velocity. The probe sends sound into the body; red blood cells reflect it back with a frequency shift proportional to how fast the blood is moving.
Students often confuse the moving-observer and moving-source formulas. They are not interchangeable: a moving observer adds or subtracts v_o in the numerator, while a moving source adds or subtracts v_s in the denominator. The asymmetry matters.
A frequent error is thinking that the Doppler effect changes the actual frequency the source emits. It does not. The source frequency stays the same; only the frequency perceived by the observer changes.
Students sometimes assume that doubling the distance halves the intensity. It does not: intensity follows the inverse-square law, so doubling the distance reduces intensity to one quarter.
Forgetting that the decibel scale is logarithmic leads to arithmetic errors. An increase from 60 dB to 70 dB is a tenfold increase in intensity, not a small step.
The four Doppler scenarios (moving observer approaching/receding, moving source approaching/receding) are almost certain to appear. Know which formula to use for each and whether the observed frequency goes up or down.
Expect a calculation question combining the decibel formula with the inverse-square law: given the intensity at one distance, find the decibel level or the intensity at another distance.
The echo/reflection formula is often tested in the context of a moving car or a bat using echolocation.
A conceptual question may ask: "Does the Doppler effect change the speed of the wave?" The answer is no. It changes only the observed frequency (and wavelength). Wave speed is set by the medium.
True or false: Tripling your distance from a point source reduces the sound intensity to one ninth.
True. Intensity follows the inverse-square law: I is proportional to 1/r squared.
Fill in the blank: An increase of 10 dB corresponds to a ______ increase in intensity.
Tenfold (factor of 10).
True or false: When a source moves toward you, the observed frequency decreases.
False. It increases.
Fill in the blank: The Doppler effect changes the observed ______ but not the ______ of the wave.
Frequency; speed.
True or false: A moving observer uses the same Doppler formula as a moving source.
False. The formulas are structurally different (numerator adjustment vs denominator adjustment).
Q: A speaker emits sound with a power of 0.50 W uniformly in all directions. What is the intensity at 4.0 m from the speaker?
A: I = P / (4 pi r squared) = 0.50 / (4 pi (4.0) squared) = 0.50 / (201.1) = 0.0025 W/m squared, or about 2.5 x 10 to the negative 3 W/m squared.
Q: The intensity at a certain point is 1.0 x 10 to the negative 5 W/m squared. What is the sound level in decibels?
A: beta = 10 log_10(I / I_0) = 10 log_10(1.0 x 10^-5 / 1.0 x 10^-12) = 10 log_10(10^7) = 10 x 7 = 70 dB.
Q: An ambulance siren emits sound at 800 Hz. The ambulance approaches you at 30 m/s. The speed of sound is 343 m/s. What frequency do you hear?
A: Moving source approaching: f = f_s / (1 - v_s/v_w) = 800 / (1 - 30/343) = 800 / (313/343) = 800 x (343/313) = 877 Hz (approximately).
Q: You drive toward a stationary siren at 25 m/s. The siren emits at 500 Hz. What frequency do you hear? (v_w = 343 m/s)
A: Moving observer approaching: f = (1 + v_o/v_w) f_s = (1 + 25/343) x 500 = (368/343) x 500 = 536 Hz (approximately).
Q: A bat emits a 40,000 Hz pulse toward a moth flying away at 5 m/s. What is the approximate frequency shift of the echo? (v_w = 343 m/s)
A: delta f = -2 f_s (v_o / v_w) = -2 x 40000 x (5/343) = -1166 Hz. The echo frequency is about 38,834 Hz. The negative sign indicates the reflected frequency is lower because the moth moves away.
The Doppler effect connects to astronomy: the red shift of light from distant galaxies is an electromagnetic analogue of the acoustic Doppler effect, and it is the primary evidence that the universe is expanding.
The inverse-square law for sound intensity is the same mathematical pattern that governs gravitational and electrostatic field strengths (covered later in the course). Recognising the 1/r squared pattern here will make those topics feel familiar.
Sound intensity and decibels also feed into any later discussion of noise control, acoustics or hearing physiology in applied-science courses.
sound intensity, decibel, dB, inverse-square law, power, point source, spherical wave, plane wave, Doppler effect, Doppler shift, moving observer, moving source, approaching, receding, echo, reflection of sound, frequency shift, radar gun, ultrasound, sonar, red shift, threshold of hearing, I_0, beta, General Physics Chapter 15, University of Washington physics