Light and Sound wave problems

JM2107

I have a few questions that I need assistance understanding and solving. I would greatly appreciate any help provided to me.

1. A musical instrument is invented that consists of a can with length L and diameter L/6. The top of the can is cut out, and a string is stretched across this open end of the can. The tension in the string is adjusted so that the frequency of the fifth harmonic for the longitudinal sound in the air column in the can equals the frequency of the sixth harmonic for transverse waves on the string.

- What is the mathematical relationship between the speed of the transverse waves on the string and the speed of sound waves in the air?

- What happens to the sound produced by the instrument if the tension in the string is increased by a factor of four? (I would appreciate more in depth comment on this particular question)

2. A convex spherical mirror with a focal length of magnitude 24.0cm is placed 16.0cm to the left of a plane mirror. An object 0.80cm tall is placed midway between the surface of the plane mirror and the vertex of the spherical mirror. The spherical mirror forms multiple images of the object.

- Where are the two images of the object formed by the spherical mirror that are closest to this mirror, and how tall is each of them? Are the images real or virtual, inverted or upright?

3. A luminous object is 5.2m from a wall. You are to use a concave mirror to project an inverted image of the object on the wall, with the image 4.00 times the size of the object.

- How far should the mirror be from the wall, and what should its radius of curvature be and is the image real or virtual?

Once again, I appreciate feedback/responses from anyone that can help me. Just to add; I have tried to solve number 1 in this post and got some figures for the two speeds. I tried to solve the second question but when I realized that the light wave keeps bouncing off the mirrors I was sort of at a loss as to how to completely answer the questions. So mainly I am asking for more focus to be put on the second question for the first problem and the second and third problems as a whole if it isn’t asking for too much. Thanks for the help once again.

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Tom Mattson

Staff Emeritus
Gold Member
JM2107,

Before giving assistance, I like to see how you started these problems. I am going to give some very general comments, and get specific only after I see what you have done.

Originally posted by JM2107
1. A musical instrument is invented that consists of a can with length L and diameter L/6. The top of the can is cut out, and a string is stretched across this open end of the can. The tension in the string is adjusted so that the frequency of the fifth harmonic for the longitudinal sound in the air column in the can equals the frequency of the sixth harmonic for transverse waves on the string.

- What is the mathematical relationship between the speed of the transverse waves on the string and the speed of sound waves in the air?
In both cases, the speed v of the wave is related to the frequency f and the wavelength &lambda; by:

v=f&lambda;

You are explcitly given a relation between the two frequencies, and you can look up the speed of sound in air.

- What happens to the sound produced by the instrument if the tension in the string is increased by a factor of four? (I would appreciate more in depth comment on this particular question)
There should be a relation in your book relating wave speed to tension and linear mass density.

2. A convex spherical mirror with a focal length of magnitude 24.0cm is placed 16.0cm to the left of a plane mirror. An object 0.80cm tall is placed midway between the surface of the plane mirror and the vertex of the spherical mirror. The spherical mirror forms multiple images of the object.

- Where are the two images of the object formed by the spherical mirror that are closest to this mirror, and how tall is each of them? Are the images real or virtual, inverted or upright?

3. A luminous object is 5.2m from a wall. You are to use a concave mirror to project an inverted image of the object on the wall, with the image 4.00 times the size of the object.

- How far should the mirror be from the wall, and what should its radius of curvature be and is the image real or virtual?
Try drawing the ray diagrams and setting up the equations. We will help you solve and interpret their solutions properly, but you have to give us something to help you with.

Thanks,

JM2107

To get more specific

I have solved all of the problems with the exception of one. The question is

What happens to the sound produced by the created instrument if the tensin in the string is increased by a factor of four?

This is what I think but I am convinced that it is not right: If the force of the tension of the string in the instrument is increased by a factor of four, then the velocity of the wave on the string will ultimatly end up doubling; thereby also meaning that the frequency of the wave would double as well. The wavelength, however, will not increase but remain constant and the amplitude of the wave will deacrease, by some factor, as a direct result of the tension increase in the string. This means that the pitch of the frequency would double.

Could you please tell me whether I am right or not, and if I am wrong, could you help me to understand what really goes on in this senario.

Last edited by a moderator:

Tom Mattson

Staff Emeritus
Gold Member
Yes, that is correct. The wave speed v is related to the tension &tau; and the linear mass density &mu; by the following relation:

v=(&tau;/&mu;)1/2

Since you are not changing the length or the mass, &mu; does not change. So simply increasing the tension by 4 doubles the velocity. Since the wavelength does not change, the frequency must also double.

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