Sound Waves (HSC SSCE Physics): Revision Notes

Beats and the Doppler Effect

Introduction

The wave nature of sound creates some fascinating phenomena that we experience in everyday life. Two important effects that arise from the properties of sound waves are beats and the Doppler effect. Both of these rely on fundamental wave principles and have practical applications in music, transportation, and many other areas.

The superposition of sound producing beats

What are beats?

When two sound sources produce tones at slightly different frequencies at the same time, something interesting happens. The sound waves from both sources overlap and combine through a process called superposition. As the waves superpose, they alternately reinforce each other (constructive interference) and cancel each other out (destructive interference). This creates a pulsing or throbbing effect in the sound intensity that we hear.

This pulsing phenomenon is called beats, and the rate at which the sound grows and shrinks is called the beat frequency.

The diagram shows two sound waves with slightly different frequencies (shown in cyan at the top). When these waves combine, the resultant wave (shown below) has an amplitude that rises and falls periodically. The dashed envelope curve shows how the overall sound intensity varies with time, creating the characteristic "beating" pattern.

Understanding beat frequency

The beat frequency is remarkably simple to calculate. It equals the difference between the two source frequencies:

$f_{\text{beat}} = |f_2 - f_1|$

where $f_2$ and $f_1$ are the frequencies of the two sound sources.

Real-world applications

Beats are extremely useful in practical situations:

• Musical instrument tuning: Experienced musicians and piano tuners use beats to tune their instruments precisely. When two strings are slightly out of tune, beats can be heard. As the tuner adjusts one string closer to the correct pitch, the beat frequency decreases. When the beating stops completely, the strings are perfectly in tune.
• Aircraft and boats: Beats can sometimes be heard from aircraft or boats with two engines running at slightly different speeds. While many people might not consciously notice this effect, it becomes quite obvious once you know what to listen for.

Worked example: calculating beat frequency

Investigation: observing beats in sound

This practical investigation allows you to observe and measure beats directly.

Aim: To observe beats in sound

Materials:

• 2 signal generators with speakers (or phone apps with external speakers)

Safety considerations:

Risk	Management Strategy
Electrical equipment near water	Keep all devices plugged into 240 V mains power well away from water
Other hazards	Assess and document any additional risks in your laboratory setting

Method:

1. Set each signal generator to a frequency of $400 \text{ Hz}$. Listen carefully to the resulting sound produced.
2. Now set the first signal generator to $399 \text{ Hz}$ while keeping the second at $400 \text{ Hz}$. Listen and observe the sound created.
3. Change the frequency of the first signal generator to $398 \text{ Hz}$. Listen and observe again.
4. Set the signal generators' frequencies to different values a few Hz apart. Listen and observe the beating effect.
5. Measure the beat frequency by counting or tapping every peak in sound intensity over a period of $10$ seconds, then divide by $10$. This averaging technique reduces measurement error.
6. Change the signal generators' frequencies so they are more than $20 \text{ Hz}$ apart and listen carefully.

Results: Record your observations for each frequency setting used.

Analysis questions:

1. How did the beat frequency relate to the difference between the two source frequencies?
   • Expected finding: The beat frequency should equal the absolute difference between the two source frequencies.
2. Describe what was heard as the difference between the two source frequencies increased to beyond $20 \text{ Hz}$.
   • Expected finding: When the frequency difference exceeds approximately $20 \text{ Hz}$, beats are no longer heard. Instead, the ear perceives two separate tones. This is because $20 \text{ Hz}$ is roughly the limit of the human ear's ability to perceive amplitude variations.

Discussion points:

1. Using a diagram, explain the cause of the beats you observed. Consider how constructive and destructive interference create the pulsing pattern.
2. Explain why beat frequencies are not heard when the difference between source frequencies exceeds $20 \text{ Hz}$. This relates to the temporal resolution of human hearing.

The Doppler effect

What is the Doppler effect?

Have you ever noticed how the pitch of an ambulance siren seems to drop suddenly as it passes you? This is the Doppler effect in action. The Doppler effect is the change in observed frequency of a sound when there is relative motion between the sound source and the observer.

The physics behind the effect

The Doppler effect occurs because relative motion changes the time between successive sound wave compressions reaching the observer.

When the source approaches the observer:

• The source "chases" the sound waves it emits
• Successive compressions reach the observer sooner than if there were no motion
• The wavelength of the sound decreases
• The observed frequency increases (higher pitch)

When the source moves away from the observer:

• It takes longer for successive compressions to reach the observer
• The wavelength increases
• The observed frequency decreases (lower pitch)

This diagram shows how a moving siren creates the Doppler effect. On the right side (where the siren is moving towards the observer), the wavelengths are compressed and closer together, resulting in a higher pitch. On the left side (where the siren is moving away), the wavelengths are extended and farther apart, resulting in a lower pitch. The wavelength as emitted from the siren itself is shown in the centre.

Real-world examples

The Doppler effect can be observed in many everyday situations:

• Emergency vehicle sirens: The classic example of an ambulance or fire truck approaching and passing you
• Racing motorcycles: The distinctive change in engine note as they pass spectators
• Aircraft flyovers: Particularly noticeable with low-altitude, high-speed aircraft

The mathematics of the Doppler effect

The change in observed frequency depends on the velocities of both the source and the observer. There are several different scenarios to consider:

The table above summarises the four main cases and their formulas. The key principle is:

• When the distance between source and observer is decreasing, $f' > f$ (frequency increases, higher pitch)
• When the distance is increasing, $f' < f$ (frequency decreases, lower pitch)

General Doppler formula

When both the source and observer may be moving, we can use a general formula:

$f' = f\frac{(v + v_{\text{observer}})}{(v - v_{\text{source}})}$

where:

• $f'$ is the observed frequency
• $f$ is the source frequency
• $v$ is the speed of sound in air ($340 \text{ m} \cdot \text{s}^{-1}$ at room temperature)
• $v_{\text{observer}}$ is the velocity of the observer
• $v_{\text{source}}$ is the velocity of the source

Worked example: ambulance and car approaching

Sonic booms and supersonic motion

An interesting and extreme consequence of the Doppler effect occurs when a source moves faster than the speed of sound. In this case:

• The source overtakes the sound waves it is emitting
• No sound is heard by an observer until after the source passes
• The source creates a shock wave called a sonic boom

Remember!