Math for practice: 1. A transverse sine wave of amplitude 10 cm and wavelength 200 cm travels from left to right along a long horizontal string with a speed 100cm/s. Take the origin at the left end. At time t=0. the left end of the string is at the origin and is moving downward i. Find the frequency, angular frequency, and propagation constant 10.5 Hz, 3.14/s 0.031/cm] ii. What is the equation of the wave? [y=10sin((pi x)/(100)-pi t)] iii. What is the equation of motion of the left end of the string? [y=-10sin(pi t)]

Let's go through each part of the problem step by step.<br /><br />### i. Finding the Frequency, Angular Frequency, and Propagation Constant<br /><br />Given:<br />- Amplitude \( A = 10 \) cm<br />- Wavelength \( \lambda = 200 \) cm<br />- Speed \( v = 100 \) cm/s<br />- Time \( t = 0 \) with the left end moving downward<br /><br />**Frequency (f):**<br />The frequency is given by:<br />\[ f = \frac{v}{\lambda} = \frac{100 \text{ cm/s}}{200 \text{ cm}} = 0.5 \text{ Hz} \]<br /><br />**Angular Frequency (ω):**<br />The angular frequency is given by:<br />\[ \omega = 2\pi f = 2\pi \times 0.5 = \pi \text{ rad/s} \]<br /><br />**Propagation Constant (k):**<br />The propagation constant is given by:<br />\[ k = \frac{2\pi}{\lambda} = \frac{2\pi}{200} = \frac{\pi}{100} \text{ rad/cm} \]<br /><br />So, the values are:<br />- Frequency \( f = 0.5 \text{ Hz} \)<br />- Angular Frequency \( \omega = \pi \text{ rad/s} \)<br />- Propagation Constant \( k = \frac{\pi}{100} \text{ rad/cm} \)<br /><br />### ii. Equation of the Wave<br /><br />The general form of a sine wave is:<br />\[ y(x, t) = A \sin(kx - \omega t + \phi) \]<br /><br />Given that at \( t = 0 \), the left end (origin) is at the origin and moving downward, we can set the phase constant \( \phi = 0 \).<br /><br />Thus, the equation of the wave is:<br />\[ y(x, t) = 10 \sin\left(\frac{\pi x}{100} - \pi t\right) \]<br /><br />### iii. Equation of Motion of the Left End of the String<br /><br />The left end of the string is at the origin (x = 0). Therefore, the equation of motion for the left end is:<br />\[ y(0, t) = 10 \sin(-\pi t) = -10 \sin(\pi t) \]<br /><br />So, the equation of motion of the left end of the string is:<br />\[ y = -10 \sin(\pi t) \]<br /><br />In summary:<br />- The frequency is \( 0.5 \text{ Hz} \).<br />- The angular frequency is \( \pi \text{ rad/s} \).<br />- The propagation constant is \( \frac{\pi}{100} \text{ rad/cm} \).<br />- The equation of the wave is \( y(x, t) = 10 \sin\left(\frac{\pi x}{100} - \pi t\right) \).<br />- The equation of motion of the left end of the string is \( y = -10 \sin(\pi t) \).