Problem 48 - University Physics with Modern Physics

Question

A loudspeaker with a diaphragm that vibrates at 960 Hz is traveling at 80.0 m/s directly toward a pair of holes in a very large wall. The speed of sound in the region is 344 m/s. Far from the wall, you observe that the sound coming through the openings first cancels at \(\pm11.4^\circ\) with respect to the direction in which the speaker is moving. (a) How far apart are the two openings? (b) At what angles would the sound first cancel if the source stopped moving?

Step-by-Step Solution

Answer

(a) Openings are approximately 0.688 m apart. (b) Sound cancels near \( \pm12.5^\circ \) if the source is stationary.

Step 1: Understanding the Problem

We have a source moving towards two openings at 80.0 m/s with a sound frequency of 960 Hz, and sound travels at 344 m/s. The interference creates cancellation at angles of \( \pm11.4^\circ \). We need to find (a) the distance between the openings and (b) the angles where cancellation occurs if the source stops moving.

Step 2: Calculate the Wavelength of the Sound

The wavelength \( \lambda \) of sound when the source is moving is found using the formula \( \lambda = \frac{v_s}{f'} \) where \( v_s = 344 \text{ m/s} \) and \( f' = f \left( \frac{v_s}{v_s - v_o} \right) \). Here \( f = 960 \text{ Hz} \) and \( v_o = 80 \text{ m/s} \). Calculate \( \lambda \).

Step 3: Find the Angle for First Minimum

The path difference for cancellation at \( \pm11.4^\circ \) is \( \lambda/2 \) for the first minimum. We use \( d \sin \theta = \frac{\lambda}{2} \) to find \( d \), where \( d \) is the distance between the openings and \( \theta = 11.4^\circ \). Solve for \( d \).

Step 4: Calculate Wavelength when Source is Stationary

When the source stops, the wavelength is \( \lambda = \frac{v_s}{f} \). Calculate this new wavelength using \( v_s = 344 \text{ m/s} \) and \( f = 960 \text{ Hz} \).

Step 5: Calculate angles with stationary source

Use the formula \( d \sin \theta = \frac{\lambda}{2} \) with the stationary wavelength to solve for \( \theta \) for the first cancellation where \( d \) is the same value found in Step 3.

Key Concepts

Sound Interference

When sound waves interact, they can cause a phenomenon called sound interference. This is the result of two or more sound waves overlapping, which can alter the way we perceive sound. There are two main types of interference:

• Constructive Interference: This occurs when sound waves align in such a way that their amplitudes add together, making the sound louder.
• Destructive Interference: In this case, the sound waves are out of phase, meaning their amplitudes can cancel each other out, resulting in diminished or completely canceled sound.

Sound interference is a crucial aspect when studying acoustics, as it can greatly affect the clarity and perception of sound at various points around a source. In the context of our exercise, the interference from the loudspeaker creates specific angles where the sound cancels, indicating a point of destructive interference.

Wave Cancellation

Wave cancellation is an interesting process where two sound waves meet. If they have equal amplitude but are exactly out of phase, they can completely cancel each other out. This results in silence at certain points. In our loudspeaker example, wave cancellation causes silence at angles of \( \pm 11.4^\circ \).

The degree of cancellation depends on how perfectly out of phase the waves are. Moreover, different frequencies and speeds can affect where and when cancellation happens. When designing room acoustics or speaker setups, understanding wave cancellation helps avoid unwanted silent zones. This means arranging speakers or adjusting frequencies to ensure the best sound delivery.

In practical terms, sound engineers can use wave cancellation concepts to shape sound in concert halls or theaters to enhance acoustics or create soundproofing.

Wavelength Calculation

Wavelength is a fundamental concept that dictates the behavior of waves, including sound waves. It is the distance between successive peaks of a wave. Calculating wavelength is essential for determining how sound behaves as it interacts with the environment. The formula used for wavelength \[ \lambda = \frac{v_s}{f'} \]is adjusted according to the source's motion. Here \(v_s\) is the speed of sound, and \(f'\) is the apparent frequency when the source is moving towards the observer. This adjustment accounts for the Doppler Effect.

The Doppler Effect describes how sound frequency changes based on the motion of the source or observer, which in turn affects wavelength. When the source moves towards an observer, the frequency increases, and the wavelength decreases. Conversely, when it moves away, the frequency decreases, and the wavelength increases.

Accurate wavelength calculations help in predicting where interference patterns like wave cancellations will occur. It's also crucial in technology utilizing sound waves, such as ultrasound imaging or sonar systems.

Source Motion Effects

The movement of a sound source significantly changes how we perceive sound. This phenomenon is largely explained by the Doppler Effect. When a source moves, it changes the frequency of sound reaching an observer:

• Towards Observer: The frequency appears higher, and the wavelength is shorter due to compressed wave fronts.
• Away from Observer: The frequency seems lower, with longer wavelengths due to stretched wave fronts.

In the exercise, the loudspeaker travels at 80 m/s, altering the frequency and consequently affecting the angles at which sound cancels. When the source is at rest, only the natural frequency without Doppler shift is considered.

Understanding these effects is vital, particularly in fields like astronomy, where the Doppler Effect is used to determine the movement of stars and galaxies. In everyday applications, it's noticed when a passing car changes pitch as it zooms by, demonstrating the practical significance of these principles in observing and utilizing sound.