Mechanical Waves Solved Examples I

1.One end of the rope is attached to a pole while the other end is moved up and down with frequency of 2 Hz. Waves along the rope are spread with speed of 4 m/s. Determine the: 

a)wavelength of these waves and 
b) what is maximum speed of oscillating rope particle if oscillating amplitude is 4 cm and it type of harmonic oscillation.

$$\begin{align} & f=2Hz \\ & v=4\text{ m/s} \\ & \underline{A=5\text{ cm}} \\ & \text{a)}\lambda =? \\ & v=\lambda f \\ & \lambda =\frac{v}{f}=\frac{4}{2}=2\text{ m}\text{,} \\ & \text{b) }u=A\omega \cos \left( \omega t+{{\theta }_{0}} \right) \\ & {{\theta }_{0}}=0, \\ & f=\frac{1}{T}\Rightarrow T=\frac{1}{2}=0.2\text{ s} \\ & \omega =\frac{2\pi }{T}=4\pi \\ & u=0.05\cdot 4\pi \cos \left( 0 \right)=0.63\text{ m/s} \\ \end{align}$$
2.Mechanical wave is spreading with a speed of 300 m/s. Frequency of the wave is 200 Hz. What is the distance between two particles which have same phases? 
$$\begin{align} & v=300\text{ m/s} \\ & \underline{f=200\text{ Hz}} \\ & \lambda =? \\ & v=\lambda f \\ & \lambda =\frac{v}{f}=\frac{300}{200}=1.5\text{ m} \\ \end{align}$$
3.The sea waves hit the rock 12 times in one minute. The speed of waves is 6 m/s. Determine the wavelength of waves.

$$\begin{align} & n=12 \\ & t=60\text{ s} \\ & \underline{v=6\text{ m/s}} \\ & T=?\lambda =? \\ & T=\frac{60}{12}=5\text{ s} \\ & \lambda =v\cdot T=6\cdot 5=30\text{ m} \\ \end{align}$$

4. The man who is on anchored boat estimated that the distance between two consecutive wave hills is 6m. The boat was lifted 10 times during one minute by the wave hills. Determine the speed of spreading waves. What is the maximum speed of boat oscillations if the distance between highest and the lowest wave point is 2 m?

$$\begin{align} & \lambda =6m \\ & n=10 \\ & t=1\min =60s \\ & \underline{2A=2m} \\ & v=?,u=? \\ & \lambda =vT \\ & T=\frac{t}{n}=\frac{60}{10}=6s \\ & v=\frac{\lambda }{T}=\frac{6}{6}=1\text{ m/s} \\ & A=1\text{ m} \\ & \omega =\frac{2\pi }{T}=1.047\text{ rad/s} \\ & u=A\omega \cos \left( \omega t \right)=1\cdot 1.047\cos \left( 0 \right) \\ & u=1.047\approx 1.05\text{ m/s} \\ \end{align}$$

5.Dolphins are emitting ultrasonic sound waves of frequency 300 Hz. What is wavelength of these waves in water and what in the air? The speed of spreading sound waves in the water is 1500 m/s, and in air is 330 m/s. Is frequency changed when passing from the water waves in the air?

$$\begin{align} & f=300\text{ kHz}=300000\text{ Hz} \\ & {{v}_{water}}=1500\text{ m/s} \\ & \underline{{{v}_{air}}=330\text{ m/s}} \\ & {{\lambda }_{water}}=\frac{{{v}_{water}}}{f}=\frac{1500}{300000}=0.005\text{ m} \\ & {{\lambda }_{air}}=\frac{{{v}_{air}}}{f}=\frac{330}{300000}=0.0011\text{ m} \\ \end{align}$$

6. Transverse wave is spreading with the speed of 15 m/s along the long wire. Period of wire particle oscillation is 1.2 s. What is difference in the phase between two particles which are located at intervals of 20 m and 30 m from the wave source? 

$$\begin{align} & v=15m/s \\ & T=1.2s \\ & {{x}_{1}}=20m \\ & \underline{{{x}_{2}}=30m} \\ & \lambda =vT=15\cdot 1.2=18m \\ & \Delta x={{x}_{2}}-{{x}_{1}}=10m \\ & \Delta \varphi =\frac{2\pi }{\lambda }\Delta x=\frac{2\pi }{18}10=\frac{10\pi }{9}=200{}^\circ \\ \end{align}$$

7.Harmonic wave is spreading with the speed of 65 m/s. Two points of the medium in which the wave is spreading are at distance of 62.5 cm and they oscillate with difference in phase of a) Determine the wavelength of wave and b) determine the frequency? 

$$\begin{align} & v=65m/s \\ & \Delta x=62.5cm=0.625m \\ & \underline{\Delta \varphi =\pi /4rad} \\ & a)\lambda =? \\ & \lambda =\frac{2\pi }{\Delta \varphi }\Delta x=\frac{2\pi }{\frac{\pi }{4}}0.625=8\cdot 0.625=5m \\ & b)\lambda =? \\ & f=\frac{v}{\lambda }=\frac{65}{5}=13Hz \\ \end{align}$$

8.Acoustic wave of frequency 200 Hz is spreading in glass with speed of 500m/s. What is a wavelength of that wave in glass? What is a wavelength of that wave in the air where speed of sound is 330m/s?

$$\begin{align} & f=200Hz \\ & {{v}_{glass}}=5000m/s \\ & \underline{{{v}_{air}}=330m/s} \\ & {{\lambda }_{glass}}=\frac{{{v}_{glass}}}{f}=\frac{5000}{200}=25m \\ & {{\lambda }_{air}}=\frac{330}{200}=1.65m \\ \end{align}$$ 9. The device consist of sound source and receiver. After transmitting the sound signal to the wall device registers echo 10 s after transmitting signal. What is the distance between wall and receiver if the speed of the sound is 340m/s?

$$\begin{align} & T=10s/2=5s \\ & \underline{v=340m/s} \\ & \lambda =vT=340\cdot 5=1700m \\ \end{align}$$