18. Waves & Sound

Wave Functions

Multiple Choice
Write the mathematical representation of the wave graphed in the following two figures.
Multiple Choice
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?
Multiple Choice
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?
Multiple Choice
The vertical displacement of a wave moving along the x-axis is given by $y (x, t) = (0.045 m) \cos ((3.4 Hz) t - (4.5 m^{- 1}) x)$ . What is the wave speed and in what direction is it traveling?
Multiple Choice
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.
Open Question
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)=2.30mm cos[(16.98 rad/m^)x+(742 rad/s)t]. Being more practical, you measure the rope to have a length of 1.35 m and a mass of 0.00338 kg. You are then asked to determine the following: (d) wave speed; (e) direction the wave is traveling;
Open Question
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)=2.30mm cos[(16.98 rad/m^)x+(742 rad/s)t]. Being more practical, you measure the rope to have a length of 1.35 m and a mass of 0.00338 kg. You are then asked to determine the following: (a) amplitude; (b) frequency; (c) wavelength; (d) wave speed; (e) direction the wave is traveling; (f) tension in the rope; (g) average power transmitted by the wave.
Open Question
A wave on a string is described by D (x,t) = (2.00 cm) ✕ sin [(12.57 rad/m)x ─ (638 rad/s) t], where x is in m and t in s. The linear density of the string is 5.00 g/m. What arec. The maximum speed of a point on the string?
Open Question
A sound wave is described by D (y,t) = (0.0200 mm) ✕ sin [(8.96 rad/m)y + (3140 rad/s)t + π/4 rad], where y is in m and t is in s.b. Along which axis is the air oscillating?
Open Question
FIGURE P16.45 is a snapshot graph at t = 0 s of a 5.0 Hz wave traveling to the left.c. Write the displacement equation for this wave.
Open Question
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.
Open Question
Show that the displacement D(x,t) = ln(ax + bt), where a and b are constants, is a solution to the wave equation. Then find an expression in terms of a and b for the wave speed.
Open Question
(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 (b) frequency.
Open Question
(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 (d) wave velocity (magnitude and direction),.
Open Question
(II) Consider the point x = 1.00 m on the cord of Example 15–6. Determine (a) the maximum acceleration of this point.