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## wave mechanics

### (12 points)3. Series 37. Year - E. acoustic thermometer

Attach a string at two points at a fixed distance $L$ and ensure it is always taut during measurement. Determine the dependence of the fundamental frequency of its oscillations on temperature.

### (3 points)1. Series 37. Year - 1. Moby Dick

Some species, such as cetaceans, navigate by echolocation. Let us assume that a cetacean emits a sound signal through a larynx located precisely between the ears at a distance a. Consider a submarine is moving at the same depth as the whale. The sound bounces off the submarine and arrives at the closer ear of the whale at time $t$ from the moment of transmission. If the time delay between the sound picked up by the right and the left ear is $\Delta t$, what is the distance and direction of the submarine?

The whale expedition got a bit out of Radka's hands.

### (3 points)5. Series 36. Year - 1. flageolet under pressure

Vojta plays the cello. He lightly places his finger on a string, tuned to the frequency $f$, at a distance of $1/n$ of its length from the head of the instrument and sounds it, hearing a tone of fundamental frequency $f_1$. He then presses the string fully against the fingerboard at the same point and sounds it again. This time the instrument produces a tone of fundamental frequency $f_2$. Determine the ratio of the frequencies $f_1/f_2$ as a function of the natural number $n$.

Vojta reminisces about the cello.

### (12 points)2. Series 36. Year - E. the loudspeaker

Measure the dependence of sound intensity emitted by your loudspeaker/mobile phone/computer on the distance from the source. Furthermore, determine the dependence of sound intensity on the settings of the output volume. Do not forget to fit the data.

Jarda cannot hear much in the back row.

### (8 points)3. Series 35. Year - 4. gentle tide

Close to the shore, the speed of sea waves is influenced by the presence of the sea bed. Assume that the speed of waves $v$ is a function of the gravity of Earth $g$ and the water depth $h$. We have $v = C g^\alpha h^\beta $. Using dimensional analysis, determine the speed of the waves as a function of the depth. Constant $C$ is dimensionless, and cannot be determined using this method.

Besides the speed of the waves, swimming Jindra is also interested in the direction of incidence of the waves. Let's define a system of coordinates, where the water surface lies in the $xy$ plane. The shoreline follows the equation $y = 0$, the ocean lies in the $y > 0$ half-plane. The water depth $h$ is given as a function of distance from the shore $h = \gamma y$, where $\gamma = \const $. On the open ocean, where the speed of the waves is constant $c$ (not influenced by the depth), plane waves are propagating at incidence angle $\theta _0$ to the $x$ axis. Find a differential equation \[\begin{equation*} \der {y}{x} = \f {f}{y} \end {equation*}\] describing the shape of the wavefront close to the shore, but do not attempt to solve it, it is far from trivial. Calculate the incidence angle of the wavefront at the shoreline.

**Bonus:** Solve the differential equation and find the shape of the wavefront close to the shore.

Jindra loves simple dimensional analysis and complicated differential equations.

### (10 points)6. Series 34. Year - S. charged chord

Assume a charged chord with linear density $\rho $, uniformly charged with linear charge density $\lambda $. The tension in the chord is $T$. It is placed in a magnetic field of constant magnitude $B$ pointing in the direction of the chord in equilibrium. Your task is to describe several aspects of the chord's oscillations. First, we want to write the appropriate wave equation. Neglect the effects of electromagnetic induction (assume the chord to be a perfect insulator; that also means the charge density does not change) and find the Lorentz force acting on an unit length of the chord for small oscilations in both directions perpendicular to the equilibrium position. Use this force to write the wave equation (which will also include the effects of the tension). Apply the Fourier substitution and determine the disperse relation in the approximation of a weak field $B$; more specifically, neglect the terms that are of higher than linear order in $\beta = \frac {\lambda B}{k \sqrt {\rho T}} \ll 1$, where $k$ is the wavenumber. Find two polarization vectors, this time neglect even the linear order of $\beta $. Now suppose that in a particular spot on the chord, we create a wave oscilating only in one specific direction. How far from the original spot will be the wave rotated by ninety degrees from the original direction?

Štěpán was nostalgically remembering the third serial task.

### (10 points)5. Series 34. Year - S. resonance and damped oscillations

- On a tense rope, waves can exist with the deflection $\f {u}{x, t}$ from the equilibrium, that satisfy the wave equation with damping

\[\begin{equation*}
\ppder {u}{t} = v^2 \ppder {u}{x} + \Gamma \pder {u}{x} ,
\end {equation*}\]
where $v$ is the phase velocity and $\Gamma $ is the coefficient of damping. Do a fourier substitution and find the dispersion relation. Solve it for the wavenumber $k$. What condition, in terms of frequency $\omega $, phase velocity $v$ and the coefficient $\Gamma $, must the waves meet in order to create nodes on the rope (i.e. points in which the rope stays in equilibrium position, but around which the rope is moving)?

- Consider a jump rope attached firmly at one end to a fixed wall. At the distance $L$ from the wall, we start moving the rope up and down to create waves. The jump rope has a linear density $\lambda $ and the constant tension $T$ in the direction away from the wall. The deflection then satisfies the equation

\[\begin{equation*} \ppder {u}{t} = \frac {T}{\lambda } \ppder {u}{x} . \end {equation*}\] For the deflection of the end of the rope that is moving satisfies $\f {u_0}{t} = A \f {\cos }{\omega _0 t}$. Assume the solution can be written in the form of two planar waves moving in the opposite direction to each other. Find the solution using only the parameters given in this problem statement, that is $T$, $\lambda $, $L$, $A$ and $\omega _0$. For certain frequencies, the solution has a diverging amplitude (i.e. growing beyond any limits). Find their values and the respective wavelenghts.

Štěpán was playing with a jump rope.

### (10 points)2. Series 34. Year - 5. magnetic non-stationarities detector

The electrical circuit shown in the figure can serve as a non-stationary magnetic field detector. It consists of nine edges of a cube formed by electric wire. The electrical resistance of one edge is $R$. If this construction lies in a non-stationary homogeneous magnetic field, which has, for simplicity, a constant direction, and its magnitude changes slowly, then there are currents $I_1$, $I_2$, $I_3$ flowing at the marked spots. With the knowledge of these currents, determine the direction of the magnetic field in space and also the dependence of its magnitude on time.

Vašek thought that an electromagnetic induction problem would be welcome.

### (3 points)3. Series 33. Year - 2. …boom

A jet fighter has flown directly overhead at constant velocity parallel to the ground. We have heard a sonic boom at $t=1{,}50 \mathrm{s}$ after that, when the fighter has been at zenith distance $\theta =30.0\dg $. Find out the height of the figther above the ground.

**Bonus:** Also find the direction from which we have heard the boom relative to the place where we have seen it.

Dodo \uv {is looking forward} to aviation days.

### (12 points)1. Series 33. Year - E. bottled

How does the frequency of the sound made by blowing over a glass bottle depend on the volume of the liquid in the bottle? Discuss also the influence of the shape of the bottle on this frequency.

Legolas can't play any instrument, so he is playing hell on us