18. Waves & Sound

Write the mathematical representation of the wave graphed in the following two figures.

A transverse wave is represented by the following function: $y=\left(18\operatorname{cm}\right)\sin\left[2\pi\left(\frac{x}{2\operatorname{cm}}-\frac{t}{5s}+\frac14\right)\right]$. What is the phase angle of this wave?

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 transverse harmonic wave moving to the left has a wavelength of 2.5 m and a wave speed of 12 m/s. The amplitude of the wave is 0.1 m. At x = 0 and t = 0, the displacement of the wave is y = 0. Write the wave function for this wave.

A fellow student with a mathematical bent tells you that the wave function of a traveling wave on a thin rope is $y(x,t)=\left(2.30\operatorname{mm)}\cos[\left(16.98\text{ }rad/m\right)x+(742\text{ }rad/s\right)t]$. Being more practical, you measure the rope to have a length of $1.35\text{ m}$ and a mass of $0.00338\operatorname{kg}$. You are then asked to determine the following: (d) wave speed; (e) direction the wave is traveling;

A fellow student with a mathematical bent tells you that the wave function of a traveling wave on a thin rope is $y(x,t)=\left(2.30\operatorname{mm)}\cos[\left(16.98\text{ }rad/m\right)x+(742\text{ }rad/s\right)t]$. Being more practical, you measure the rope to have a length of $1.35\text{ m}$ and a mass of $0.00338\operatorname{kg}$. You are then asked to determine the following: (a) amplitude; (b) frequency; (c) wavelength.

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?

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.

(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.

(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).

Consider the point x = 1.00 m on the cord of Example 15–6. Determine the maximum acceleration of this point.

(I) A transverse wave on a wire is given by D (x, t) = 0.015 sin (35x - 1200 t) where D and x are in meters and t is in seconds. Write an expression for a wave with the same amplitude, wavelength, and frequency but traveling in the opposite direction.

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.

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?