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## wave mechanics

### (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

### (12 points)4. Series 32. Year - E. paper isolation

Measure the shielding of the sound by paper. As an experimental tool, you can use mobile phones as a sound generator and microphone in the computer as a sound detector (Audacity). Use papers of various kinds and shapes.

Michal wanted to know how to get rid of unpleasant sounds emitted by his roommate.

### (7 points)1. Series 32. Year - 4. Skyfall

When James Bond let go of agent 006 Alec Treveljan from the top of the Arecibo radiotelescope in the final scene of the film Golden Eye, the falling agent started screaming with a frequency $f$. How does the frequency agent 007 hears at the top of the telescope change as a function of time. Neglect air resistance.

**Hint:** Ask Mr. Doppler

Matej enjoys looking outside

### (7 points)3. Series 30. Year - 3. where's the whistle

Verca's ears can be aproximated by two point detectors separated by distance $d$, which can detect incoming sound waves equally well from all directions. Verca can determine the location of a known source extremely well and so, one day, just as she woke up, she asked her friends to test her. However, Verca forgot an earplug in one ear, reducing the intensity in her left ear $ktimes$. Verca was blindfolded and a source was placed at a position $y$ in front of her and $x$ to her right (or $-x$ to her left). Determine the position ( $x′,y′)$ Verca will point to if she determines the position of the source using the intensity of the sound.

Lubosek got frightened by a phone while wearing a single headphone.

### (2 points)5. Series 28. Year - 2. I hear well, I can't say

At a distance $d=5\;\mathrm{m}$ from a point-like source of sound we hear a noise of the level of intensity $L_{1}=90dB$. At what distance from the source of the sound is the level of intensity of the sound $L_{2}=50dB?$

Karel wanted to have something from accoustics here again.

### (5 points)3. Series 28. Year - P. whistle me something

Explain the principle upon which whistling with your mouth works. Consider first simple models and gradually transfer to more complicated ones. Then Choose the best and on their basis determine the range within which the base frequency can be.(If you know how to whistle you can determine the accuracy of your estimate.)

Mirek wants to inconspicously find out how many others also don't know how to whistle.

### (6 points)3. Series 27. Year - S. Aplicational

- In the text of the seriesy we used the approximative relation √( 1 + $h)$, where $his$ a small value. Determine the precision of such an approximation. How much can $h$ differ from zero so that the approximated value and the precise one shall differ only by 10%? We can make a similar approximation for any „normal“ (read occuring in nature) function using Taylor's series expansion. Try to find the Tylor's series of cos$h$ and sin$h$ on the internet and neglect factors with a higher order than $h$ and find the approximate border value where it differs by approximately 0.1.

- Considering a wave equation for a classical string from the serial and let the string be fastened on one end in the point [ $x;y]=[0;0]$ a na druhém konci v bodě [ $x;y]=[l;0]$. For what values of $ω,α,aabis$ the expression

$$y(x,t)=\sin ({\alpha} x)\left [a\sin {({\omega} t)} b\cos {({\omega} t)}\right ]$$

a solution of the wave equation?
*Tip* Subsitute into the equation for motion and use the boundary conditions.

- In the previous part of the series we were comparing different values of action for different trajectories of different particles. Now calculate the value of Nambu-Gota's action for a closed string which from time 0 to time $t$ stands still un the plane ( $x¹,x)$ and has the shape of a circle with radius $R$. Thus we have

$$X({\tau} , {\sigma} )=(c{\tau} , R\cos {{\sigma} }, R\sin {{\sigma} },0)$$

for $τ∈\langle0,2π\rangle$. Furthermore sketch the worldsheet of this string (forget about the last zero component) and how the line of a constant $τ$ and $σ$ look.

### (4 points)4. Series 26. Year - 4. Hit it with a hammer

Imagine hitting one end of an iron rod and observing the resulting sound waves. Describe (using drawings) the time dependence of the wavefronts in the plane of the rod. We are especially interested in what the wavefronts would look like at two particular moments. The first one is the time when the sound wave reaches the other end of the rod, and the second one is the moment the wave reaches the original end of the rod after reflecting at the other end. Do not forget to describe how did you construct your drawings. You can assume that there are only longitudinal oscillations of the rod, and that its diameter is negligable compared to its length. The ratio of the speed of sound in the rod and in the air is $β=v_{rod}⁄v_{air}≈10$.

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