18. Waves & Sound

If the speed of a wave ($v$) increases while its wavelength ($\lambda$) remains constant, what happens to the wave's frequency ($f$)?

The function for some transverse wave is ? = (0.5 m) sin [(0.8 m⁻¹)x − 2?(50 Hz)t + π/3]. What is the transverse velocity at t=2 s, x=7 cm? What is the maximum transverse speed? The maximum transverse acceleration?

A horizontal string tied at both ends is vibrating in its fundamental mode. The traveling waves have speed $v$, frequency $f$, amplitude $A$, and wavelength $\(\lambda\)$. How much time does it take the string to go from its largest upward displacement to its largest downward displacement at the points located at (i) $x = λ/2$, (ii) $x = λ/4$, and (iii) $x = λ/8$, from the left-hand end of the string.

(II) A transverse traveling wave on a cord is represented by D = 0.22 sin (5.6x + 34 t) where D and x are in meters and t is in seconds. For this wave determine wave velocity (magnitude and direction).

A sound wave is described by $D(y,t)=\left(0.0200\text{mm}\right)\times \sin \left[\right(8.96\text{rad/m})y+(3140\text{rad/s})t+\pi /4\text{rad}]$, where $y$ is in $m$ and $t$ is in $s$. Along which axis is the air oscillating?

Write the equation for the wave in Problem 29 traveling to the right, if its amplitude is 0.020 cm, and D = - 0.020 cm, at t = 0 and x = 0.

FIGURE EX16.8 is a picture at t = 0 s of the particles in a medium as a longitudinal wave is passing through. The equilibrium spacing between the particles is 1.0 cm. Draw the snapshot graph D(x, t = 0 s) of this wave at t = 0 s.

(II) A transverse traveling wave on a cord is represented by D = 0.22 sin (5.6x + 34 t) where D and x are in meters and t is in seconds. For this wave determine frequency.

A horizontal string tied at both ends is vibrating in its fundamental mode. The traveling waves have speed $v$, frequency $f$, amplitude $A$, and wavelength $\lambda$. What is the amplitude of the motion at the points located at (i) $x=\lambda /2$, (ii) $x=\lambda /4$, and (iii) $x=\lambda /8$, from the left-hand end of the string?

A transverse wave on a cord is given by D(x,t) = 0.12 sin (3.0x - 15.0t), where D and x are in meters and t is in seconds. At t = 0.20s, what are the displacement, velocity, and acceleration of a point on the cord where x = 0.60 m?

Show that the displacement D(x,t) = cx² + dt², where c and d are constants, is a solution to the wave equation. Then find an expression in terms of c and d for the wave speed.

A transverse wave is represented by the following function: $y=\(\left\)(18\(\operatorname{cm}\]\right\))\(\sin\[\left\)[2\(\pi\]\left\)(\(\frac{x}{2\operatorname{cm}\)}-\(\frac{t}{5s}\)+\(\frac\)14\(\right\))\(\right\)]$. What is the phase angle of this wave?

A transverse wave pulse travels to the right along a string with a speed v = 2.0 m/s. At the shape of the pulse is given by the function D = 0.45 cos (2.6x + 1.2), where D and x are in meters. Determine a formula for the wave pulse at any time t assuming there are no frictional losses.