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Wagnerian Opera. A man marries a great Wagnerian soprano but, alas, he discovers he cannot stand Wagnerian opera. In order to save his eardrums, the unhappy man decides he must silence his larklike wife for good. His plan is to tie her to the front of his car and send car and soprano speeding toward a brick wall. This soprano is quite shrewd, however, having studied physics in her student days at the music conservatory. She realizes that this wall has a resonant frequency of 600 \(\mathrm{Hz}\) , which means that if a continuous sound wave of this frequency hits the wall, it will fall down, and she will be saved to sing more Isoldes. The car is heading toward the wall at a high speed of 30 \(\mathrm{m} / \mathrm{s}\) . (a) At what frequency must the soprano sing so that the will will crumble? (b) What frequency will the soprano hear reflected from the wall just before it crumbles?

Step by step solution

01

Understand the Doppler Effect

The problem involves sound waves and relative motion, which is an application of the Doppler effect. The Doppler effect describes the change in frequency of a wave in relation to an observer moving relative to the source of the wave.

02

Set the Given Parameters

We have the car moving towards the wall, which means the observer (wall) is stationary, and the source (soprano) is moving towards the wall. The given speed of the car is 30 m/s, and the resonant frequency of the wall is 600 Hz. The speed of sound in air is typically around 343 m/s.

03

Use Doppler Effect Formula for Moving Source

The formula for the observed frequency \( f' \) when the source is moving towards a stationary observer is:\[ f' = \frac{f}{1 - \frac{v_s}{v}} \]where:- \( f \) is the frequency of the source (soprano must sing at this frequency).- \( v_s \) is the speed of the source (30 m/s).- \( v \) is the speed of sound (343 m/s).- \( f' \) is the observed frequency (600 Hz, since the wall should resonate at this frequency).

04

Solve for the Soprano's Singing Frequency

Rearrange the equation to solve for \( f \):\[ f = f' \times (1 - \frac{v_s}{v}) = 600 \times (1 - \frac{30}{343}) \]Calculate \( f \).

05

Calculate Frequency to be Sung

Substituting in the values:\[ f = 600 \times \left(1 - \frac{30}{343}\right) = 600 \times \left(1 - 0.0875\right) = 600 \times 0.9125 \approx 547.5 \space \mathrm{Hz} \]Thus, the soprano must sing at approximately 547.5 Hz.

06

Find Frequency Heard By Soprano

To find the frequency the soprano hears reflected from the wall, apply the Doppler effect as the wall becomes the source and the soprano the observer moving towards it:\[ f'' = f' \times \left(1 + \frac{v_s}{v}\right) = 600 \times (1 + \frac{30}{343}) \]Calculate \( f'' \).

07

Calculate the Reflected Frequency

Substituting in the values:\[ f'' = 600 \times \left(1 + \frac{30}{343}\right) = 600 \times \left(1 + 0.0875\right) = 600 \times 1.0875 \approx 652.5 \space \mathrm{Hz} \]Thus, the soprano hears a frequency of approximately 652.5 Hz reflected back.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Resonant Frequency

Resonant frequency is a natural frequency at which an object, like a wall, vibrates with maximum amplitude. In the context of the opera scenario, the wall's resonant frequency is 600 Hz. This means that if the soprano hits exactly this frequency while moving towards the wall, she can cause it to vibrate intensely.

Objects have resonant frequencies because they contain internal structures that can be prompted to enter sympathetic vibrations. When external sound waves match this frequency, the object absorbs more sound energy which leads to larger vibrations.

• Components of the sound wave match the structure's natural mode.
• Maximum energy transfer occurs, resulting in high amplitude oscillations.
• For the wall in our problem, this could result in enough movement to cause it to crumble.

Speed of Sound

The speed of sound is the speed at which sound waves travel through a medium. In air, this speed is approximately 343 meters per second (m/s) at room temperature.

Sound waves require a medium to travel because they propagate through the particles in the medium. This speed can change based on:

• Temperature: Warmer temperatures generally increase the speed of sound.
• Humble Composition: The mix of gases (like air) affects how fast the sound can travel.

In the opera problem, the speed of sound is a crucial factor. It affects how quickly the sound reaches the wall and how modifications in frequency are perceived due to the Doppler effect.

Frequency of Sound Waves

Frequency of sound waves is the number of complete wave cycles passing a point per second, measured in Hertz (Hz). The frequency determines the pitch of the sound.

For the soprano in our scenario, the frequency she sings is measured just like any other sound. However, its interplay with motion due to the Doppler effect changes how it's observed.

• Frequency affects how sound is received or emitted by moving objects.
• In her case, 547.5 Hz is the frequency she must aim for to cause the wall to resonate at 600 Hz when her motion is accounted for.

Relative Motion in Waves

Relative motion in waves is described by the Doppler effect. This effect occurs when there is relative motion between the source of waves and an observer. It leads to a perceived shift in frequency.

In this problem:

• The soprano is moving forward, towards the wall, which alters the perceived frequency of her singing by the stationary wall.
• The formula applied here explains how the frequency the wall "hears" increases due to her motion, which is different from her actual singing frequency.
• The Doppler effect also explains why the reflected frequency back to the soprano is higher than what she originally sang.

Simply put, movement towards a listener compresses sound waves, increasing frequency, while moving away decreases it.