Question on waves and vibrations

[the_godfather]

Homework Statement

this is part of a problem sheet and is the last question i have left to work out. i have been given solutions of 105dB and 86dB.

A person is in a room in which two sources of sound of loudness 100 dB and
98 dB are emitting the same frequency. The sources form a standing wave pattern in the room. What are the maximum and minimum loudness the person will be subjected to as he moves about?

Homework Equations

The Attempt at a Solution

not to sure i understand the question. i think it asks what is the constructive and destructive superposition that can occur. if so, what is my next step
i'm guessing that the Standing wave ratio formula is a part of this but what is the amplitude reflection co-eff?

thanks

 

[cepheid]

If the two sound waves interfere entirely constructively, their amplitudes (intensities) will simply add together. If they interfere entirely destructively, their amplitudes (intensities) will subtract. Therefore, these are the extremal conditions, i.e. the represent the maximum and minimum possible sound intensities the person can hear. Of course, first you have to figure out what the individual amplitudes are, then add or subtract them, then see what the result is in dB.

 

[kerimek]

I would approach this question by first understanding the concepts of waves and vibrations. In this scenario, we have two sources of sound emitting the same frequency, which means they are in phase and will create a standing wave pattern in the room. A standing wave is formed when two waves with the same frequency and amplitude travel in opposite directions and interfere with each other, resulting in stationary points (nodes) and points of maximum displacement (antinodes).

The loudness of a sound is measured in decibels (dB) and is a relative measure of the intensity of the sound. In this case, the person in the room will experience different loudness levels as they move about due to the constructive and destructive interference of the two sound sources. The maximum loudness will occur at the antinodes, where the two waves are in phase and reinforce each other, resulting in a louder sound. The minimum loudness will occur at the nodes, where the two waves are out of phase and cancel each other out, resulting in a quieter sound.

To calculate the maximum and minimum loudness, we can use the formula for sound intensity level, which is measured in dB and is given by:

β = 10log(I/I0)

Where β is the sound intensity level, I is the sound intensity, and I0 is the reference intensity (usually taken as 10^-12 W/m^2).

Since we are given the loudness levels of the two sound sources (100 dB and 98 dB), we can use the above equation to calculate the corresponding sound intensities (I1 and I2). Then, using the principle of superposition, we can add these two intensities to determine the maximum and minimum loudness levels experienced by the person in the room.

The next step would be to calculate the amplitude reflection coefficient, which is a measure of how much of the incident sound is reflected at the boundary of the room. This will depend on the properties of the room and can be calculated using the room's dimensions, material properties, and the frequency of the sound. Once we have this coefficient, we can use the standing wave ratio formula to determine the constructive and destructive interference that occurs at different locations in the room.

In conclusion, the maximum and minimum loudness levels experienced by the person in the room can be calculated using the principles of sound intensity level, superposition, and standing wave ratio. Understanding these concepts and using the appropriate formulas will help