The wave equation and its relationship with SHM

The equation $y(x, t) = A \cdot \sen(k \cdot x - \omega \cdot t)$ 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.

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.

In a laboratory, a transverse wave is generated in a taut string whose equation, in SI units, is $y(x, t) = 0,05 \cdot \sen(4 \cdot \pi \cdot x - 20 \cdot \pi \cdot t)$. 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 transverse wave travels along a string according to the equation, in SI units, $y(x, t) = 0,03 \cdot \sen(5 \cdot x + 20 \cdot t)$.

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.

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.

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 $\Delta \varphi = \frac{2 \cdot \pi}{\lambda} \cdot \Delta x$.
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.

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 longitudinal wave propagates along a taut helical spring with equation, in SI units, $y(x, t) = 0,04 \cdot \sen(2 \cdot \pi \cdot x - 10 \cdot \pi \cdot t)$.

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

A steel cable in a vibration test bench has a linear density $\mu = 4\cdot10^{-3}\ \text{kg/m}$ 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.

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.