Help with questions on Sound Waves and Harmoics

[kim3k84]

1.) A transverse wave is traveling with a speed of 300m/s on a horizontal string. If the tension in the string is increased by a factor of four, what is the speed of the wave?

2.) A wire has a cross-sectional area of 4.2 X 10-8 m2 and is made from a material whose density is 7900 kg/m3. Determine the wire's linear density (m/L).

3.) A uniform cord has a mass of 0.300 kg and a total length of 6.00 m. Tension is maintained in the cord by suspending a 2.00 kg mass from one end. Find the speed of a pulse on this cord. Assume the tension is not affected by the mass of the cord.

4.) A middle C string of the C-major scale on a piano has a fundamental frequency of 262 Hz. And the A note has a fundamental frequency of 440 Hz.

a.) Calculate the frequencies of the next two harmonics of the C string.

b.) If the strings for the A and C notes are assumed to have the same mass per unit length and the same length, determine the ratio of tensions in the two strings.

c.) In a real piano, the assumption we made in (b) is only half true. The string densities are equal, but the A string is 64% as long as the C string. What is the ratio of their tensions?

 

[Borek]

You just don't ask. You try, you tell us what you did, we push you in the right direction.

 

[nlightner]

1.) When the tension in the string is increased by a factor of four, the speed of the wave will also increase by a factor of four. This is because the speed of a transverse wave on a string is directly proportional to the tension in the string. Therefore, the new speed of the wave will be 1200m/s.

2.) Linear density (m/L) is a measure of the mass per unit length of a material. It can be calculated by dividing the density (ρ) of the material by its cross-sectional area (A). Therefore, the linear density of the wire can be calculated as:

m/L = ρ/A = (7900 kg/m3) / (4.2 X 10-8 m2) = 1.88 X 105 kg/m

3.) The speed of a pulse on a cord is determined by the tension in the cord and the mass per unit length of the cord. In this scenario, the tension is maintained by a 2.00 kg mass, which is suspended from one end of the cord. Therefore, the tension in the cord can be calculated as:

T = mg = (2.00 kg) (9.8 m/s2) = 19.6 N

The mass per unit length of the cord can be calculated as:

μ = m/L = (0.300 kg) / (6.00 m) = 0.05 kg/m

Using the formula for the speed of a wave on a string, v = √(T/μ), we can calculate the speed of the pulse on the cord as:

v = √(19.6 N / 0.05 kg/m) = 19.8 m/s

4.) a.) The next two harmonics of the C string will have frequencies that are multiples of the fundamental frequency. The second harmonic will have a frequency of 2 x 262 Hz = 524 Hz. The third harmonic will have a frequency of 3 x 262 Hz = 786 Hz.

b.) The ratio of tensions in the two strings can be calculated using the formula for the frequency of a vibrating string, f = √(T/μL), where T is the tension, μ is the mass per unit length, and L is the length of the string. Since the strings have the same length and mass per unit length, the ratio of their tensions can be calculated as:

T(A)/