# Sound Waves and the Doppler Effect

Physics I · Principles of Waves, Fluids and Thermodynamics · https://tryals.app/learn/physics-i/sound-waves-and-the-doppler-effect

## Sound Waves and Properties

The siren's pitch drops the moment the ambulance passes; wavefronts bunch ahead of a moving source and stretch behind it.

**Sound** is a longitudinal pressure wave propagating through a medium via particle compressions and rarefactions.

The **speed of sound** depends on the elastic and inertial properties of the medium:

$$v = \sqrt{\frac{B}{\rho}}, \qquad v = 331 + 0.6T_C \text{ m/s (air)}$$

where $B$ is bulk modulus, $\rho$ is density, and $T_C$ is temperature in Celsius.

**Sound intensity level** ($\beta$) is measured on a logarithmic decibel scale relative to the threshold of hearing:

$$\beta = 10\log_{10}\left(\frac{I}{I_0}\right), \quad I_0 = 10^{-12}\text{ W/m}^2$$

| Sound Source | Intensity Level |
| --- | --- |
| Threshold of hearing | $0$ dB |
| Normal conversation | $60$ dB |
| Rock concert | $110$ dB |
| Pain threshold | $130$ dB |

## The Doppler Effect

The **Doppler effect** occurs when a source or observer moves, causing a shift in **observed frequency** ($f'$):

$$f' = f\left(\frac{v \pm v_o}{v \mp v_s}\right)$$

where $v$ is wave speed, $v_o$ is observer speed, and $v_s$ is source speed. Use top signs for approaching, bottom for receding.

| Motion Type | Wavefront Effect | Frequency Shift |
| --- | --- | --- |
| **Approaching** | Bunch together | Higher ($f' > f$) |
| **Receding** | Stretch apart | Lower ($f' < f$) |

> **Common pitfall:** For a source at constant speed, pitch does not slide continuously. You hear a steady high pitch approaching, a steady low pitch receding, and an abrupt drop at passing.

> **Applications:** Radar speed guns, medical ultrasound, and cosmic redshift ($z = \Delta\lambda/\lambda$), proving the universe expands.

## Practice questions

7 of this lesson's 13 practice questions, with answers. The full set is in the app.

### 1. Order these milestones in the history of acoustics.

**Answer:**

1. Pythagoras studies musical string ratios
2. Mersenne measures the speed of sound in air
3. Doppler publishes the Doppler principle
4. Mach defines the ratio of speed to sound speed
5. First sonar systems developed for submarine detection

**Why:** Acoustics history: Pythagoras (c. 500 BCE), Mersenne (1636), Doppler (1842), Mach (1887), sonar (c. 1915).

Page: https://tryals.app/practice/physics-i/sound-waves-and-the-doppler-effect/order-these-milestones-in-the-history-of-acoustics

### 2. An ambulance drives **past** you at constant speed with its siren on. Which best describes the pitch you hear?

A. Steadily higher while approaching, lower after passing
B. Constant, but louder as it gets closer
C. Higher while approaching only if it is also accelerating
D. Rising continuously as it approaches, falling continuously as it recedes

**Answer:** A. Steadily higher while approaching, lower after passing

**Why:** Constant velocity gives a constant Doppler shift: one fixed higher pitch inbound, one fixed lower pitch outbound, with a rapid slide only during the moment of passing. The familiar "eee-yooo" is two plateaus and a step, not a long glissando.

Page: https://tryals.app/practice/physics-i/sound-waves-and-the-doppler-effect/an-ambulance-drives-past-you-at-constant-speed-with-its-siren-on

### 3. You see a lightning flash and hear the thunder **6 seconds** later. Estimate how far away the strike was, in kilometres (sound: ~340 m/s).

**Answer:** 2 (within ±0.4)

**Why:** About 2 km, the origin of the "3 seconds per kilometre" storm rule. The trick works because light outruns sound by a factor of nearly a million.

Page: https://tryals.app/practice/physics-i/sound-waves-and-the-doppler-effect/you-see-a-lightning-flash-and-hear-the-thunder-6-seconds-later

### 4. Sound waves in air are transverse waves: air molecules oscillate perpendicular to the direction of travel.

**Answer:** False

**Why:** False. Sound in air is **longitudinal**: molecules oscillate parallel to the travel direction, producing compressions and rarefactions. Fluids cannot sustain shear, so they carry no transverse waves at all; that is why seismic S-waves stop at the Earth's liquid outer core, which is how we know it is liquid.

Page: https://tryals.app/practice/physics-i/sound-waves-and-the-doppler-effect/sound-waves-in-air-are-transverse-waves-air-molecules-oscillate

### 5. A source emits sound at $500 Hz$ and moves toward a stationary observer at $34 m/s$. Taking the speed of sound as $340 m/s$, the observed frequency is approximately ___ Hz. **Set the slider to your answer.**

**Answer:** 555.6 (within ±20)

**Why:** $f_{obs} = f \times v/(v - v_s) = 500 \times 340/(340 - 34) = 500 \times 340/306 \approx 555.6$ Hz.

Page: https://tryals.app/practice/physics-i/sound-waves-and-the-doppler-effect/a-source-emits-sound-at-500-hz-and-moves-toward-a-stationary-observer

### 6. The Doppler shift differs depending on whether the source or the observer is moving, even at the same relative speed. Why does the medium break the symmetry?

A. Because the two cases give identical results and the difference is apparent only
B. Because the source’s motion changes the frequency it emits
C. Because the speed is fixed relative to the AIR, not the source
D. Because the observer’s ears respond differently to approaching sound

**Answer:** C. Because the speed is fixed relative to the AIR, not the source

**Why:** A medium provides a preferred frame, so "who is moving" is a real question. For light there is no such frame and only relative velocity appears, which is precisely the puzzle special relativity resolved.

Page: https://tryals.app/practice/physics-i/sound-waves-and-the-doppler-effect/the-doppler-shift-differs-depending-on-whether-the-source-or-the

### 7. Which statements about the Doppler effect are correct?

A. Observed frequency decreases when source moves away
B. Observed frequency increases when source approaches observer
C. Effect is absent when source and observer both move at the same velocity
D. The Doppler effect applies only to sound, not light

**Answer:** A. Observed frequency decreases when source moves away; B. Observed frequency increases when source approaches observer; C. Effect is absent when source and observer both move at the same velocity

**Why:** The Doppler effect applies to all waves including light. Relative motion is what matters; equal velocities $\to$ no shift.

Page: https://tryals.app/practice/physics-i/sound-waves-and-the-doppler-effect/which-statements-about-the-doppler-effect-are-correct