16.6 Standing waves and resonance  (Page 10/17)

A transverse wave on a horizontal string $\left(\mu =0.0060\text{kg/m}\right)$ is described with the equation $y\left(x,t\right)=0.30\text{m}\text{sin}\left(\frac{2\pi }{4.00\text{m}}\left(x-v_{w}t\right)\right).$ The string is under a tension of 300.00 N. What are the wave speed, wave number, and angular frequency of the wave?

A student holds an inexpensive sonic range finder and uses the range finder to find the distance to the wall. The sonic range finder emits a sound wave. The sound wave reflects off the wall and returns to the range finder. The round trip takes 0.012 s. The range finder was calibrated for use at room temperature $T=20\text{°}\text{C}$ , but the temperature in the room is actually $T=23\text{°}\text{C}.$ Assuming that the timing mechanism is perfect, what percentage of error can the student expect due to the calibration?

A wave on a string is driven by a string vibrator, which oscillates at a frequency of 100.00 Hz and an amplitude of 1.00 cm. The string vibrator operates at a voltage of 12.00 V and a current of 0.20 A. The power consumed by the string vibrator is $P=IV$ . Assume that the string vibrator is $90\text{\%}$ efficient at converting electrical energy into the energy associated with the vibrations of the string. The string is 3.00 m long, and is under a tension of 60.00 N. What is the linear mass density of the string?

A traveling wave on a string is modeled by the wave equation $y\left(x,t\right)=3.00\text{cm}\text{sin}\left(8.00{\text{m}}^{\mathrm{-1}}x+100.00{\text{s}}^{\mathrm{-1}}t\right).$ The string is under a tension of 50.00 N and has a linear mass density of $\mu =0.008\text{kg/m}.$ What is the average power transferred by the wave on the string?

A transverse wave on a string has a wavelength of 5.0 m, a period of 0.02 s, and an amplitude of 1.5 cm. The average power transferred by the wave is 5.00 W. What is the tension in the string?

(a) What is the intensity of a laser beam used to burn away cancerous tissue that, when $90.0\text{\%}$ absorbed, puts 500 J of energy into a circular spot 2.00 mm in diameter in 4.00 s? (b) Discuss how this intensity compares to the average intensity of sunlight (about) and the implications that would have if the laser beam entered your eye. Note how your answer depends on the time duration of the exposure.

Consider two periodic wave functions, $y_1\left(x,t\right)=A\text{sin}\left(kx-\omega t\right)$ and $y_2\left(x,t\right)=A\text{sin}\left(kx-\omega t+\varphi \right).$ (a) For what values of $\varphi$ will the wave that results from a superposition of the wave functions have an amplitude of 2 A ? (b) For what values of $\varphi$ will the wave that results from a superposition of the wave functions have an amplitude of zero?

Consider two periodic wave functions, $y_1\left(x,t\right)=A\text{sin}\left(kx-\omega t\right)$ and $y_2\left(x,t\right)=A\text{cos}\left(kx-\omega t+\varphi \right)$ . (a) For what values of $\varphi$ will the wave that results from a superposition of the wave functions have an amplitude of 2 A ? (b) For what values of $\varphi$ will the wave that results from a superposition of the wave functions have an amplitude of zero?

A trough with dimensions 10.00 meters by 0.10 meters by 0.10 meters is partially filled with water. Small-amplitude surface water waves are produced from both ends of the trough by paddles oscillating in simple harmonic motion. The height of the water waves are modeled with two sinusoidal wave equations, $y_1\left(x,t\right)=0.3\text{m}\text{sin}\left(4{\text{m}}^{\mathrm{-1}}x-3{\text{s}}^{\mathrm{-1}}t\right)$ and $y_2\left(x,t\right)=0.3\text{m}\text{cos}\left(4{\text{m}}^{\mathrm{-1}}x+3{\text{s}}^{\mathrm{-1}}t-\frac{\pi }{2}\right).$ What is the wave function of the resulting wave after the waves reach one another and before they reach the end of the trough (i.e., assume that there are only two waves in the trough and ignore reflections)? Use a spreadsheet to check your results. ( Hint: Use the trig identities $\text{sin}\left(u\pm v\right)=\text{sin}u\text{cos}v\pm \text{cos}u\text{sin}v$ and $\text{cos}\left(u\pm v\right)=\text{cos}u\text{cos}v\mp \text{sin}u\text{sin}v)$