16.5 Interference of Waves

A string of a length of 2.00 m with a linear mass density of $\mu = 0.006 \text{kg}/\text{m}$ is attached to the end of a 2.00-m-long string with a linear mass density of $\mu = 0.012 \text{kg}/\text{m} .$ The free end of the higher-density string is fixed to the wall, and a student holds the free end of the low-density string, keeping the tension constant in both strings. The student sends a pulse down the string. Describe what happens at the interface betwee…

Consider two sinusoidal sine waves traveling along a string, modeled as $y_{1} \left(x , t\right) = 0.3 \text{m} \text{sin} \left(4 \text{m}^{−1} x + 3 \text{s}^{−1} t + \frac{\pi}{3}\right)$ and $y_{2} \left(x , t\right) = 0.6 \text{m} \text{sin} \left(8 \text{m}^{−1} x - 6 \text{s}^{−1} t\right) .$ What is the height of the resultant wave formed by the interference of the two waves at the position $x = 1.0 \text{m}$ at time $t = 3.0 \text{s}?$

Two sinusoidal waves are moving through a medium in the same direction, both having amplitudes of 3.00 cm, a wavelength of 5.20 m, and a period of 6.52 s, but one has a phase shift of an angle $\phi$. What is the phase shift if the resultant wave has an amplitude of 5.00 cm? [Hint: Use the trig identity $\text{sin} u + \text{sin} v = 2 \text{sin} \left(\frac{u + v}{2}\right) \text{cos} \left(\frac{u - v}{2}\right)$

Two sinusoidal waves are moving through a medium in the positive x-direction, both having amplitudes of 7.00 cm, a wave number of $k = 3.00 \text{m}^{−1} ,$ an angular frequency of $\omega = 2.50 \text{s}^{−1} ,$ and a period of 6.00 s, but one has a phase shift of an angle $\phi = \frac{\pi}{12} \text{rad} .$ What is the height of the resultant wave at a time $t = 2.00 \text{s}$ and a position $x = 0.53 \text{m}?$

Two sinusoidal waves, which are identical except for a phase shift, travel along in the same direction. The wave equation of the resultant wave is $y_{R} \left(x , t\right) = 0.70 \text{m} \text{sin} \left(\right. 3.00 \text{m}^{−1} x - 6.28 \text{s}^{−1} t +$
$\pi / 16 \text{rad} \left.\right) .$ What are the angular frequency, wave number, amplitude, and phase shift of the individual waves?

Consider two wave functions that differ only by a phase shift, $y_{1} \left(x , t\right) = A \text{cos} \left(k x - \omega t\right)$ and $y_{2} \left(x , t\right) = A \text{cos} \left(k x - \omega t + \phi\right) .$ Use the trigonometric identities $\text{cos} u + \text{cos} v = 2 \text{cos} \left(\frac{u - v}{2}\right) \text{cos} \left(\frac{u + v}{2}\right)$ and $\text{cos} \left(− \theta\right) = \text{cos} \left(\theta\right)$ to find a wave equation for the wave resulting from the superposition of the two waves. Does the resulting wave function come as a surprise to you?