# Tranverse waves and velocity problem

• dmolson
In summary, a piano string with a length of 1.6 m and mass density of 26 mg/m has a fundamental frequency of 450 Hz. The speed of transverse waves on the string can be calculated using the relationship between tension, mass density, and wave speed. The wavelength and frequency of the sound wave produced by the string's vibration in air can also be determined using the speed of sound in air.
dmolson
A piano string of length 1.6 m and mass density 26 mg/m vibrates at a (fundamental) frequency of 450 Hz.

(a) What is the speed of the transverse string waves?
(b) What is the tension?
c) What are the wavelength and frequency of the sound wave in air produced by vibration of the string? The speed of sound in air at room temperature is 340 m/s.

I have been working on this problem and cannot seem to come up with the solution. Any help would be greatly appreciated.

What have you done so far?

What's the wavelength of the fundamental mode? How are speed, wavelength, and frequency related?

What's the relationship between tension, mass density, and wave speed?

(a) The speed of transverse waves on a string can be calculated using the formula v = √(T/μ), where v is the speed, T is the tension, and μ is the mass density. In this case, the mass density is given as 26 mg/m, which is equivalent to 0.026 kg/m. The frequency of the string is given as 450 Hz, which is the same as 450 vibrations per second. Using the given length of 1.6 m, we can calculate the wavelength using the formula λ = v/f, where λ is the wavelength, v is the speed, and f is the frequency. Substituting in the values, we get λ = (0.026 kg/m/450 Hz) = 0.0000578 m. This means that the speed of the transverse waves on the string is v = √(T/μ) = √(T/0.026 kg/m) = √(T/0.026) = √(T/0.026) = √(T/0.026) = 0.00904 m/s.

(b) The tension of the string can be calculated using the formula T = μv², where T is the tension, μ is the mass density, and v is the speed. Substituting in the values, we get T = (0.026 kg/m)(0.00904 m/s)² = 0.0000218 N. This is the tension required to produce the given frequency of 450 Hz on the string.

(c) The wavelength of the sound wave produced by the vibrating string can be calculated using the formula λ = v/f, where λ is the wavelength, v is the speed of sound in air at room temperature (340 m/s), and f is the frequency of the string (450 Hz). Substituting in the values, we get λ = (340 m/s)/(450 Hz) = 0.7556 m. This is the wavelength of the sound wave produced by the vibrating string. The frequency of the sound wave will be the same as the frequency of the string, which is 450 Hz.

## What is a transverse wave?

A transverse wave is a type of wave in which the particles of the medium vibrate perpendicular to the direction of propagation. An example of a transverse wave is a water wave, where the water molecules move up and down as the wave moves forward.

## What is the velocity of a transverse wave?

The velocity of a transverse wave depends on the properties of the medium it is traveling through. In a string, the velocity is equal to the square root of the tension divided by the linear density of the string. In general, the velocity of a transverse wave is given by the equation v = λf, where λ is the wavelength and f is the frequency of the wave.

## How do you calculate the wavelength of a transverse wave?

The wavelength of a transverse wave can be calculated by dividing the velocity of the wave by its frequency. In other words, the wavelength is equal to the speed of the wave divided by the number of cycles per second.

## What is the relationship between frequency and wavelength in transverse waves?

The frequency and wavelength of a transverse wave are inversely proportional to each other. This means that as the frequency increases, the wavelength decreases, and vice versa. This relationship is described by the equation f = v/λ, where f is the frequency, v is the velocity, and λ is the wavelength.

## How does the amplitude of a transverse wave affect its velocity?

The amplitude of a transverse wave does not affect its velocity. The velocity of a wave is determined by the properties of the medium it is traveling through, not its amplitude. However, a larger amplitude can result in a higher energy wave, which can cause more significant disturbances in the medium.

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