Source: PHYS 116, University of Washington, Sections 15.5–15.7
Difficulty: Introductory to Intermediate | Prerequisites: Sections 15.3 and 15.4 (wave equations, v = λf), logarithms (base 10), basic algebra.
These sections take the wave descriptions from 15.3 and 15.4 and ask: how much energy does a wave carry, and what happens to it as the wave spreads out? You will learn how intensity (power per unit area) drops with distance, how humans perceive loudness on a logarithmic scale measured in decibels, and how relative motion between a source and an observer shifts the frequency you hear or see. This is the part of the waves unit that connects the maths to things you experience directly: why an ambulance siren changes pitch as it passes, why a shout is painfully loud up close but barely audible far away, and why distant galaxies appear redder than they are.
Intensity is power per unit area (W/m²) and falls off as 1/r² for a spherical wave. Loudness is measured in decibels on a logarithmic scale, where every 10 dB increase corresponds to a tenfold increase in intensity. The Doppler effect shifts the observed frequency higher when the source and observer approach each other and lower when they move apart; for light this produces blue shift and red shift respectively.
Intensity (I)
The power delivered per unit area by a wave, measured in W/m². Think of it as how much energy the wave dumps onto each square metre of surface every second.
Wave front
A surface connecting all points of the wave at the same phase. For a point source, wave fronts are spheres (3D) or circles (2D). Very far from the source, small sections of spherical wave fronts look flat, and these are called plane waves.
Spherical wave
A wave that spreads outward from a point source equally in all directions. Its intensity drops as 1/r² because the same power is spread over an ever-larger sphere.
Decibel (dB)
The unit used to express sound intensity level on a logarithmic scale. Defined as β = (10 dB) log₁₀(I / I₀), where I₀ = 1.0 × 10⁻¹² W/m² is the threshold of hearing.
Threshold of hearing (I₀)
The lowest intensity sound that the average human ear can detect in an extremely quiet room: I₀ = 1.0 × 10⁻¹² W/m². This is the reference point for the decibel scale.
Sound intensity level (β)
The loudness of a sound expressed in decibels relative to the threshold of hearing. In simple terms, this is the number you read off a decibel meter.
Doppler effect
The change in observed frequency that occurs when a wave source and an observer move relative to each other. Think of it as the reason a siren sounds higher-pitched as it approaches and lower-pitched as it moves away.
Red shift
The shift of light to longer wavelengths (toward the red end of the visible spectrum) when a light source is receding from the observer.
Blue shift
The shift of light to shorter wavelengths (toward the blue end) when a light source is approaching the observer.
Echo
A reflected wave. When a sound wave bounces off a surface and returns, you hear the reflection as an echo.
Shock wave
A very large amplitude wave produced when a source moves faster than the wave speed in the medium (v_s > v). Think of it as a sonic boom: the source outruns its own waves, and they pile up into one massive wavefront.
The lines (or surfaces) that locate the crests of a wave are called wave fronts. They are spaced one wavelength apart.
A circular wave (2D) or spherical wave (3D) has curved wave fronts that expand outward from a point source.
Very far from the source, small sections of spherical wave fronts appear flat. These are plane waves.
Intensity is the power delivered per unit area: I = P / a, where P is power in watts and a is area in m².
SI units of intensity: W/m².
When a wave hits a surface, the power absorbed in area a is P = Ia.
A wave concentrated onto a small area has high intensity; the same power spread over a large area has low intensity.
For a spherical wave, the power spreads uniformly over the surface of a sphere of area 4πr².
Intensity of a spherical wave at distance r from the source: I = P / (4πr²).
This means intensity falls off as 1/r² (the inverse-square law).
To compare intensities at two distances without knowing the power: I₁/I₂ = r₂²/r₁², or equivalently I₂ = I₁(r₁²/r₂²).
Remember: P = ΔE/Δt (power is the rate of energy transfer).
Increasing the sound intensity by a factor of 10 results in an increase in perceived loudness by a factor of approximately 2.
Human perception of loudness is logarithmic, not linear. This is why the decibel scale uses a log.
The threshold of hearing is I₀ = 1.0 × 10⁻¹² W/m². This is the quietest sound a typical human ear can detect in a very quiet room.
Sound intensity level (in dB): β = (10 dB) log₁₀(I / I₀).
To find intensity from a given dB level: I = I₀ × 10^(β / 10 dB).
0 dB: threshold of hearing (1.0 × 10⁻¹² W/m²)
10 dB: person breathing at 3 m
20 dB: a whisper at 1 m
60 dB: normal conversation at 1 m
80 dB: vacuum cleaner
110 dB: home stereo at max volume
120 dB: rock concert
130 dB: threshold of pain
Every time the actual intensity increases by a factor of 10, the sound intensity level increases by 10 dB.
Sound is perceived as roughly twice as loud for each 10 dB increase.
A person shouting emits 1.0 W of sound energy. At 1.0 m away:
I = P / (4πr²) = 1.0 / (4π(1.0)²) = 0.080 W/m²
β = (10 dB) log₁₀(0.080 / 1.0 × 10⁻¹²) = 110 dB
When a source moves toward an observer, the wave crests in front of the source are bunched together (shorter wavelength, higher frequency).
When a source moves away from an observer, the wave crests behind the source are stretched apart (longer wavelength, lower frequency).
After a wave crest leaves the source, its motion is governed by the properties of the medium, not the source. The subscript v_s refers to the speed of the source, not the speed of the waves.
Approaching source: f₊ = f_s / (1 − v_s/v)
Receding source: f₋ = f_s / (1 + v_s/v)
If you drive toward a stationary siren, you encounter crests more frequently, so the observed frequency is higher.
Approaching observer: f₊ = (1 + v₀/v) f_s
Receding observer: f₋ = (1 − v₀/v) f_s
A light source receding from you produces a longer detected wavelength than the emitted wavelength. This is red shift (shifted toward the red end of the visible spectrum).
A light source approaching you produces a shorter detected wavelength. This is blue shift.
Reflected waves are called an echo.
The frequency shift for a reflected wave is Δf = ±2f_s (v₀/v), where v₀ is the speed of the reflecting object.
If the reflecting object is stationary (v₀ = 0), there is no frequency shift.
When the source speed v_s exceeds the wave speed v, the source outruns the waves it produces.
The amplitudes of the overlapping waves add up to produce a very large amplitude wave, called a shock wave.
Anything that moves faster than the speed of sound in air will create a shock wave (a sonic boom).
A police siren has frequency 550 Hz as it approaches and 450 Hz after it passes. The speed of sound is 343 m/s. Setting up the two Doppler equations for the approaching and receding cases and subtracting gives v_s = (f₊ − f₋)/(f₊ + f₋) × v = (100/1000) × 343 = 34 m/s.
I = \frac{P}{a}Intensity equals power divided by area.
I = \frac{P}{4\pi r^2}Intensity of a spherical wave at distance r from the source.
\frac{I_1}{I_2} = \frac{r_2^2}{r_1^2}Comparing intensities at two distances (inverse-square law).
\beta = (10 \text{ dB}) \log_{10}\!\left(\frac{I}{I_0}\right)Sound intensity level in decibels. I₀ = 1.0 × 10⁻¹² W/m².
I = I_0 \cdot 10^{\beta / 10 \text{ dB}}Intensity from a known decibel level.
f_+ = \frac{f_s}{1 - v_s / v}Doppler effect: observed frequency when source approaches (moving source).
f_- = \frac{f_s}{1 + v_s / v}Doppler effect: observed frequency when source recedes (moving source).
f_+ = \left(1 + \frac{v_0}{v}\right) f_sDoppler effect: observed frequency when observer approaches (moving observer).
f_- = \left(1 - \frac{v_0}{v}\right) f_sDoppler effect: observed frequency when observer recedes (moving observer).
\Delta f = \pm 2 f_s \frac{v_0}{v}Frequency shift for a wave reflected off a moving object.
The inverse-square law explains why you need to stand close to a speaker at a concert to feel the bass, and why doubling your distance from a siren makes it sound noticeably quieter (the intensity drops to one quarter). The decibel scale is used in occupational health regulations to set safe noise exposure limits in factories and on construction sites. The Doppler effect is the basis of police radar guns (reflected microwaves shift in frequency depending on car speed), medical ultrasound (measuring blood flow velocity) and astronomical red-shift measurements that revealed the expansion of the universe.
Students often think that doubling the distance from a source halves the intensity. It does not. Intensity follows an inverse-square law, so doubling the distance cuts the intensity to one quarter.
Students sometimes confuse intensity (a physical quantity in W/m²) with loudness (a perceptual quantity). A sound can have measurably higher intensity without sounding dramatically louder, because human perception is logarithmic.
A common error on Doppler problems is mixing up the formulas for a moving source and a moving observer. They are different equations. For a moving source, the v_s term is in the denominator; for a moving observer, the v₀ term is in the numerator.
Students sometimes assume the Doppler effect changes the speed of the wave. It does not. The wave speed is determined by the medium. Only the observed frequency (and therefore wavelength) changes.
⚠️ Be able to calculate intensity at a given distance using I = P/(4πr²) and convert between intensity and decibels in both directions.
⚠️ Know that each factor-of-10 increase in intensity adds 10 dB and roughly doubles perceived loudness.
⚠️ Doppler effect problems will test whether you can pick the right formula (moving source vs. moving observer) and get the signs correct. Approaching always raises the frequency; receding always lowers it.
⚠️ The inverse-square ratio I₁/I₂ = r₂²/r₁² lets you compare intensities at two distances without knowing the source power. This is a common exam shortcut.
⚠️ Shock waves were noted as not on the exam in the lecture notes, but understand the basic concept: source speed exceeds wave speed.
True or false: if you triple your distance from a point source, the intensity drops to one ninth.
Fill in the blank: the threshold of hearing is I₀ = ____ W/m².
True or false: a 20 dB increase in sound intensity level means the intensity has increased by a factor of 100.
Fill in the blank: when a sound source approaches you, the observed frequency is ____ (higher/lower) than the emitted frequency.
True or false: the Doppler effect changes the speed of the wave.
Q: A speaker emits 0.50 W of sound power. What is the intensity at a distance of 2.0 m?
A: I = P/(4πr²) = 0.50/(4π(2.0)²) = 0.50/50.3 ≈ 0.010 W/m².
Q: What is the sound intensity level of a sound with intensity 1.0 × 10⁻⁵ W/m²?
A: β = (10 dB) log₁₀(1.0 × 10⁻⁵ / 1.0 × 10⁻¹²) = (10 dB) log₁₀(10⁷) = 70 dB.
Q: A fire engine siren emits at 600 Hz. You are stationary and the engine approaches at 30 m/s. What frequency do you hear? (Speed of sound = 343 m/s.)
A: f₊ = f_s / (1 − v_s/v) = 600 / (1 − 30/343) = 600 / 0.913 ≈ 657 Hz.
Q: You are 4.0 m from a loudspeaker and measure an intensity of 0.020 W/m². What is the intensity at 12 m?
A: I₂ = I₁(r₁²/r₂²) = 0.020 × (4.0²/12²) = 0.020 × (16/144) = 0.0022 W/m².
Q: A police siren sounds at 550 Hz as it approaches and 450 Hz after it passes. What is the speed of the police car? (v = 343 m/s.)
A: v_s = (f₊ − f₋)/(f₊ + f₋) × v = (100/1000) × 343 = 34 m/s.
Intensity and the inverse-square law connect directly to the wave equations from Sections 15.3 and 15.4, since intensity depends on wave power and distance. The decibel scale is a practical application of logarithms, which appear again in other contexts such as the Richter scale for earthquakes. The Doppler effect for light is foundational in astronomy and cosmology (Hubble's discovery of the expanding universe relies on red-shift measurements) and connects to later topics in optics and modern physics.
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