# Wave on a String 1#

## Useful Info#

Waves on a string satisfy \(v = f \psi = w/k = \sqrt{T_s u}\) , where \(f\) is the frequency of the source, \( \psi \) is the wavelength of the sinusoidal oscillation, \(w = 2 \pi f = 2 \pi /T\) is the angular frequency, and \(k = 2 \pi / \psi \) is the wavenumber, \(T_s\) is the tension in the string, and \(u\) is the linear density (mass per unit length) of the string. The speed of a wave on a string is a property of the string (its tension and mass per unit length). Standing waves on strings are only possible at frequencies where reflections from both ends of the string reinforce to produce nodes and antinodes at particular locations. A string fixed at both ends has nodes at both ends- a displacement of zero at all times at these locations. The fundamental frequency is the lowest frequency oscillation consistent with the boundary conditions. For a given string (of known tension and mass per unit length) the velocity is fixed, so the longest wavelength (or smallest wavenumber) oscillation corresponds to the lowest or fundamental frequency.

## Part 1#

A standing wave on a string fixed at x = 0.00m and at x = 10m satisfies y(x, t) = 2Asin(kx)cos(t).

At \(x = 0.10 m\), a fixed string implies that \(y(x = 0.1,t) = 0\). What values of the wave number, \(k\), are possible? Express your answer as a formula in terms of a number, \( \pi \), and \(m\), with units of \(rad/m\).

### Answer Section#

Please enter a numeric value in \( \pi m\).

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## Part 2#

For the fundamental frequency (or first harmonic), what value of \(m\) should be chosen?

Note that the longest wavelength allowed by the boundary conditions is associated with the fundamental frequency of the oscillation on the string.

### Answer Section#

Please enter in a numeric value in \(m\).

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## Part 3#

For the fundamental frequency, what is the displacement (the value of \(y\)) of this wave at \(x = 0.05 m\) at \(t = 0 s\)? Express your answer as a formula in terms of numbers and A.

\(w = (2 \pi )/T\)

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## Part 4#

For the fundamental frequency, what is the displacement (the value of \(y\)) of this wave at \(x = 0.05 m\) at \(t = T/4\), where \(T\) is the period of the fundamental mode?

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## Part 5#

For the fundamental frequency, what type(s) of energy does this wave have at \(t = 0 s\)?

### Answer Section#

Potential energy

Kinetic energy

Both potential and kinetic energy

Neither potential nor kinetic energy

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## Part 6#

For the fundamental frequency, what type(s) of energy does this wave have at \(t = T/4 s\)?

### Answer Section#

Potential energy

Kinetic energy

Both potential and kinetic energy

Neither potential nor kinetic energy

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## Attribution#

Problem is licensed under the CC-BY-NC-SA 4.0 license.