Exercises: The wave equation and its relation to SHM

In a laboratory, a transverse wave is generated in a taut string whose equation, in SI units, is $y(x, t) = 0.05 \times \sen(4 \times \pi \times x - 20 \times \pi \times t)$. The figure shows the profile of the string at an instant.

1. a) State the values of the amplitude $A$, the wave number $k$ and the angular frequency $\omega$.
2. b) Reason, from the sign of the term in $t$ within the phase, in which direction along the $x$ axis the wave propagates.

A tuning fork vibrates and generates a harmonic wave of angular frequency $\omega = 100 \times \pi\ \text{rad/s}$ in the air.

1. a) Calculate its period $T$, remembering that $T = \frac{2 \times \pi}{\omega}$.
2. b) Calculate its frequency $f$, in $\text{Hz}$.

Waves are generated on the water surface in a laboratory ripple tank with a wavenumber $k = 2 \times \pi\ \text{rad/m}$. The figure shows the surface profile at a given instant.

1. a) Calculate the wavelength $\lambda$, remembering that $\lambda = \frac{2 \times \pi}{k}$.
2. b) Explain in your own words what the wavelength represents physically by observing the surface profile.

The human ear perceives sound between $20\ \text{Hz}$ and $20\ \text{kHz}$, and sound travels through air at $340\ \text{m/s}$.

1. a) Calculate the wavelength of the lowest-pitched sound we can hear using $v = \lambda \times f$.
2. b) Calculate the wavelength of the highest-pitched sound we can hear, first expressing $20\ \text{kHz}$ in hertz.
3. c) Reason, in light of the results, how the wavelength varies as the frequency increases if the speed remains unchanged.

The equation $y(x, t) = A \times \sen(k \times x - \omega \times t)$ describes a doubly periodic disturbance: in time and in space. The two graphs in the figure represent this double periodicity.

1. a) If we choose a fixed point in the medium (fixing $x$), with what period does it oscillate? Express it in terms of $\omega$.
2. b) If we observe the disturbance profile at a fixed instant (fixing $t$), over what distance does it repeat? Express it in terms of $k$.
3. c) Explain, based on the above, why wave motion is said to combine an oscillation in time with periodicity in space.

A longitudinal wave propagates along a taut helical spring with equation, in SI units, $y(x, t) = 0.04 \times \sen(2 \times \pi \times x - 10 \times \pi \times t)$.

1. a) Calculate its frequency $f$ and its period $T$.
2. b) Calculate its wavelength $\lambda$.
3. c) Calculate its propagation speed using $v = \frac{\omega}{k}$.

When a harmonic wave travels through a medium, every point of the medium oscillates with simple harmonic motion (SHM).

1. a) State which quantities of the SHM of each point coincide with those of the wave: consider the amplitude and the frequency.
2. b) State which quantity distinguishes the oscillation of a point near the source from that of a point further away.
3. c) Explain the difference between the speed of oscillation of a point in the medium and the propagation speed of the wave.

A transverse wave travels along a string according to the equation, in SI units, $y(x, t) = 0.03 \times \sen(5 \times x + 20 \times t)$.

1. a) Determine in which direction along the $x$-axis the wave propagates and justify your answer using the sign of the phase.
2. b) Calculate its propagation speed from $k$ and $\omega$.
3. c) Rewrite the equation making only the necessary changes so that the wave travels in the positive direction of the axis.

An oceanographic buoy records the passage of a wave train with amplitude $1.5\ \text{m}$ and period $T = 4\ \text{s}$; the distance between two consecutive crests is $25\ \text{m}$. The figure shows the wave profile.

1. a) Calculate the angular frequency $\omega$ and the wavenumber $k$.
2. b) Calculate the wave propagation speed using $v = \lambda \times f$.
3. c) Write down the wave equation assuming zero initial phase and propagation in the positive direction of the $x$-axis.

A harmonic wave propagates along a string with equation, in SI units, $y(x, t) = 0.02 \times \sen(10 \times \pi \times x - 200 \times \pi \times t)$.

1. a) Calculate the displacement of the point located at $x = 0.5\ \text{m}$ at time $t = 0.01\ \text{s}$.
2. b) State the maximum displacement this point can reach and reason whether it depends on the chosen point.
3. c) Calculate the period with which this point oscillates.

To study the phase of a sound wave of $f = 85\ \text{Hz}$ travelling through air at $340\ \text{m/s}$, two microphones are placed in line with the loudspeaker emitting it, as shown in the figure.

1. a) Calculate the wavelength of the sound.
2. b) Calculate the phase difference between the oscillations of two points separated by $1\ \text{m}$, using $\Delta \varphi = \frac{2 \times \pi}{\lambda} \times \Delta x$.
3. c) The microphones are finally placed a distance $2\ \text{m}$ apart: calculate the new phase shift and reason whether in this case the points oscillate in phase or in phase opposition.

In a geological survey, a harmonic seismic wave of amplitude $0.01\ \text{m}$ and frequency $f = 10\ \text{Hz}$ is generated, propagating through the ground at $6000\ \text{m/s}$ in the positive direction of the $x$-axis.

1. a) Calculate its wavelength and its wavenumber.
2. b) Calculate its angular frequency.
3. c) Write down its wave equation given that the initial phase is $\varphi_0 = \frac{\pi}{2}\ \text{rad}$.

A point at the centre of a violin string oscillates with amplitude $A = 5\times10^{-4}\ \text{m}$ when the string emits an A note of $f = 440\ \text{Hz}$.

1. a) Calculate the angular frequency of the oscillation.
2. b) Knowing that the maximum speed of oscillation in SHM is $v_{max} = A \times \omega$, calculate the maximum speed of that point.
3. c) The wave travels along that string at $250\ \text{m/s}$: compare this value with that from the previous section and reason why both speeds are so different even though they are measured on the same string.

A laboratory ultrasound generator emits waves of $f = 40\ \text{kHz}$ that propagate through air at $340\ \text{m/s}$.

1. a) Express the frequency in hertz and calculate the wavelength, in metres and in millimetres.
2. b) Calculate the period and express it in seconds and in microseconds ($1\ \mu\text{s} = 10^{-6}\ \text{s}$).
3. c) Reason, knowing that the human ear perceives at most about $20\ \text{kHz}$, why no one hears the pitch emitted by the generator.

A snapshot of a string shows a sinusoidal profile whose crests rise $5\ \text{cm}$ above the equilibrium position and are separated by $80\ \text{cm}$; the disturbance travels along the string at $2\ \text{m/s}$. The figure shows this snapshot.

1. a) Determine the amplitude and wavelength from the description of the snapshot, expressing them in metres.
2. b) Calculate the frequency, period, angular frequency and wavenumber.
3. c) Write down the wave equation with zero initial phase and propagation in the positive direction of the $x$-axis, with all quantities in SI units.

A transverse wave with amplitude $0.04\ \text{m}$ and period $T = 0.5\ \text{s}$ propagates along a string at $8\ \text{m/s}$ in the negative direction of the $x$-axis.

1. a) Calculate its angular frequency.
2. b) Calculate its wavelength and its wavenumber.
3. c) Write down its wave equation with zero initial phase, taking care with the sign of the phase so that it represents the specified direction of propagation.

A harmonic wave of $f = 50\ \text{Hz}$ travels along a laboratory string at $200\ \text{m/s}$. The figure shows the wave profile with three marked points P, Q and R.

1. a) Calculate its wavelength.
2. b) Calculate the minimum distance between two points on the string oscillating in phase and justify the result.
3. c) Calculate the minimum distance between two points oscillating in phase opposition.
4. d) Determine the phase difference between two points separated by $3\ \text{m}$ and describe how one oscillates relative to the other.

A transverse wave of amplitude $0.05\ \text{m}$, angular frequency $\omega = 20 \times \pi\ \text{rad/s}$ and speed $10\ \text{m/s}$ propagates along a taut string in the positive direction of the $x$-axis. At time $t = 0$, the point at the origin reaches its maximum positive displacement.

1. a) Calculate the wavenumber $k$.
2. b) Determine the initial phase $\varphi_0$ by imposing the condition $y(0, 0) = A$.
3. c) Write down the complete wave equation.
4. d) Calculate the displacement of the point $x = 1\ \text{m}$ at time $t = 0.05\ \text{s}$ and reason the sign of the result.

A steel cable in a vibration test bench has a linear density $\mu = 4\times10^{-3}\ \text{kg/m}$ and is subjected to a tension $T = 40\ \text{N}$; the speed of transverse waves in the cable is $v = \sqrt{\frac{T}{\mu}}$. A harmonic wave of amplitude $0.02\ \text{m}$ and frequency $f = 100\ \text{Hz}$ is generated, travelling in the positive direction of the $x$-axis with zero initial phase.

1. a) Calculate the propagation speed of the wave.
2. b) Calculate the wavelength, wavenumber and angular frequency.
3. c) Write down the wave equation.
4. d) If the tension were doubled while keeping the same frequency, calculate the new speed and wavelength, and reason why these quantities change while the frequency does not.

In a laboratory wave flume, a wave of amplitude $0.1\ \text{m}$ and speed $4\ \text{m/s}$ propagates in the positive direction of the $x$-axis. At time $t = 0$, the surface profile, with crests separated by $2\ \text{m}$, is given by $y(x, 0) = 0.1 \times \sen(\pi \times x)$.

1. a) Determine $k$ and $\omega$ and write down the complete wave equation.
2. b) Express the displacement of the point $x = 0.5\ \text{m}$ as a function of time and reason whether immediately after $t = 0$ this point moves up or down.
3. c) Check the initial displacement of the point $x = 1\ \text{m}$ and deduce in which direction it starts to move.

In an ultrasound scan, ultrasound waves of $f = 3.5\ \text{MHz}$ are used, propagating through soft tissue at $1540\ \text{m/s}$.

1. a) Calculate the wavelength in metres and express it in millimetres.
2. b) The echo reflected from an internal structure is received $8\times10^{-5}\ \text{s}$ after emission: taking into account that the pulse travels twice that distance, calculate the depth of the structure.
3. c) Reason why, in order to observe details of a few millimetres, it is desirable to work with megahertz frequencies rather than audible sound.

A harmonic wave of $f = 2\ \text{Hz}$ and amplitude $0.05\ \text{m}$ travels in the positive direction of the $x$-axis along a light string at $6\ \text{m/s}$ and passes into a second, denser string attached to the first, where its speed is $3\ \text{m/s}$. The frequency is conserved at the boundary between the two media. Observe the diagram of the two joined strings:

1. a) Calculate the wavelength in each string.
2. b) Write down the wave equation in each of the strings, with zero initial phase.
3. c) State which quantities are conserved and which change when passing from one string to the other.
4. d) Reason, considering how the junction between both strings oscillates, why the frequency cannot change at that point.

A transverse wave propagates along a string at $v = 50\ \text{m/s}$ with a frequency of $25\ \text{Hz}$.

1. a) Differentiate, from $y = A \times \sen(k \times x - \omega \times t)$, the expression for the oscillation speed of a point on the string and show that its maximum value is $A \times \omega$.
2. b) Calculate the amplitude the wave should have for the maximum oscillation speed of the points to equal the propagation speed.
3. c) Reason, specifying the direction and physical meaning of each, why the oscillation speed and the propagation speed are completely different physical quantities.

In a taut string, two points separated by $1\ \text{m}$ are observed to always oscillate in phase.

1. a) Justify why the wavelength can only take the values $\lambda = \frac{1}{n}\ \text{m}$, with $n = 1, 2, 3, \dots$.
2. b) If it is also known that $\lambda > 0.4\ \text{m}$, determine all possible values of $\lambda$.
3. c) The propagation speed in the string is $100\ \text{m/s}$: calculate the frequency corresponding to each possible value.
4. d) Propose an additional measurement that would allow determining which of the values is the actual one and justify your proposal.