II. Problems
1.Prove that wave function \( y(x, t)=y_{m} \sin (k x-\omega t) \) is a solution for the \( 1 \mathrm{D} \) wave equation: \( \frac{\partial^{2} y}{\partial x^{2}}=\frac{1}{v^{2}} \frac{\partial^{2} y}{\partial t^{2}} \).
2. A stationary police car uses a detector, which emits microwave with frequency \( 3.0 \times 10^{10} \mathrm{~Hz} \), to detect the speeds of cars on a highway. If a reflected wave, when combined with the outgoing wave, produces beats at a rate of \( 5000 \mathrm{~s}^{-1} \). If the speed limit in the highway is \( 70 \mathrm{~km} / \mathrm{hr} \), is the car exceeding speed limit?
3. Suppose an \( \mathrm{S} \) wave passes through a boundary in rock where its speed decrease from \( 5.0 \mathrm{~km} / \mathrm{s} \) to \( 4.0 \mathrm{~km} / \mathrm{s} \). If the angle of incidence at the boundary is \( 40.0^{\circ} \), what is the angle of refraction?.
4. At a distance of \( 30 \mathrm{~m} \) from a jet engine, the sound intensity level is 120 \( \mathrm{dB} \). At what distance will the intensity level decreases to \( 100 \mathrm{~dB} \) ? Assume the engine is an isotropic source of sound and ignore reflections and absorption.
. 5.A tuning fork \( (600 \mathrm{~Hz}) \) can be used to tune a piano wire with the same frequency. What fraction in the tension of the piano wire needs to increase when the occurrence of 8.0 beats/s between the fork and the piano wire is detected?
6. An elastic string of mass \( m=1.0 \mathrm{~g} \) and length \( L=1.0 \mathrm{~m} \) has a tension \( F= \) \( 100 \mathrm{~N} \). What are the wavelengths and frequencies of the first two harmonic modes?
