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A stationary source emits sound waves of frequency \(f_{\mathrm{s}}\). There is no wind blowing. A device for detecting sound waves and measuring their observed frequency moves toward the source with speed \(v_{\mathrm{L}}\) and the observed frequency of the sound waves is \(f_{\mathrm{L}}\). The measurement is repeated for different values of \(v_{\mathrm{L}}\). You plot the results as \(f_{\mathrm{L}}\) versus \(v_{\mathrm{L}}\) and find that your data lie close to a straight line that has slope \(1.75 \mathrm{~m}^{-1}\) and \(y\) -intercept \(600.0 \mathrm{~Hz}\). What are your experimental results for the speed of sound in the still air and for the frequency \(f_{\mathrm{s}}\) of the source?

Step by step solution

01

Identify the formula

The Doppler effect for sound waves emitted by a stationary source, where a detector moves toward the source in a still medium, can be expressed using the formula:\(f_L = f_s\left(1 + \frac{{v_L}}{{v_s}}\right)\)where \(f_s\) is the frequency of the source, \(v_s\) is the speed of sound in the still air, \(v_L\) is the speed of the detector, and \(f_L\) is the observed frequency.

02

Solve for the speed of sound

By expressing \(f_L\) in terms of \(v_L\) (by finding the equation of the line on the graph), the equation becomes:\(f_L = f_s\frac{{v_L}}{{v_s}} + f_s\)The slope of the equation given in the problem is \(1.75 \mathrm{~m}^{-1}\), and the y-intercept is \(600.0 \mathrm{~Hz}\), so\(f_s\frac{1}{{v_s}} = 1.75 \mathrm{~m}^{-1}\)Isolating \(v_s\) to one side gives:\(v_s = \frac{1}{1.75 \mathrm{~m}^{-1}} = 0.571 \mathrm{~m/s} = 571 \mathrm{~m/s}\)

03

Solve for the source frequency

We can substitute the value of \(v_s\) into the second part of our line equation to find \(f_s\):\(f_s = 600.0 \mathrm{~Hz}\), which is the y-intercept of our plotted graph.

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Key Concepts

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

Sound Waves Frequency

Sound waves are an example of pressure waves and occur as a result of vibrations in a medium, which could be solid, liquid or gas. They have a characteristic known as frequency, denoted by the symbol f, which is the number of waves that pass a certain point in one second, measured in Hertz (Hz). The frequency determines the pitch of the sound; higher frequencies produce a higher-pitched sound and vice versa. It's essential to understand that the frequency emitted by a source (like a siren or a musical instrument) is constant in a uniform medium, like still air, unless the source or observer is moving.

Speed of Sound in Air

The speed at which sound waves travel through a medium, particularly air, is vital for understanding phenomena such as the Doppler effect. In air, this speed is affected by factors like temperature, humidity, and air pressure, but at room temperature (approximately 20°C), the speed of sound in air is about 343 meters per second (m/s). This value plays a crucial role in solving problems related to sound wave propagation, and when it's known, one can make accurate calculations about the behavior of sound waves in that medium.

Observed Frequency

The observed frequency, or the frequency perceived by a listener, can differ from the source's frequency based on the relative motion between the observer and the source. This is primarily due to the Doppler effect. When the observer is moving towards the source, the observed frequency increases; conversely, it decreases when moving away from the source.

• If the observer is stationary, and the source moves, the effect also takes place, albeit differently based on whether the source is moving towards or away from the observer.
• If there is no relative motion, the observed frequency remains the same as the source's frequency.

Understanding this concept is essential for interpreting situations where the motion of the observer alters the frequency detected. This can be in everyday scenarios like an ambulance passing by or in more complex systems such as radars and medical imaging technologies.

Doppler Effect Formula

The Doppler effect formula models the relationship between the observed frequency and the source frequency taking into account the motion of the source and/or the observer. The basic formula when the source is stationary, and the observer is moving towards the source is:
\[f_L = f_s\left(1 + \frac{{v_L}}{{v_s}}\right)\]
where:

• \(f_s\) is the frequency produced by the source,
• \(v_s\) represents the speed of sound in the medium,
• \(v_L\) is the velocity of the listener moving towards the source, and
• \(f_L\) denotes the observed frequency by the listener.

From this formula, if the slope of the observed frequency to the listener's speed is plotted, it reflects the ratio of the source frequency over the speed of sound. This relationship allows us to solve for either variable when the other is known, and it is especially useful in experiments and calculations that involve sound wave phenomena.