Wave Functions Practice Problems

During a laboratory experiment, a group of students observes the formation of a progressive wave along a string using a simple harmonic oscillator. The string has a length of 1.2 m and a mass of 5 g. The progressive wave function y(x, t) is given by y = 8 mm (sin {[3π (rad/cm) x] - [33π (rad/s) t] } ). The teacher asks the student to find out the i) speed (v), ii) and direction of propagation of the progressive wave.

A student uses a frequency generator and an oscillator to produce a progressive wave along a string.  The string has a length of 1.2 m and a mass of 5 g. The produced wave is modeled with the wave function y(x,t) = 12 mm sin([π/2(rad/m) x] - [100π (rad/s) t]). Determine the progressive wave's i) amplitude (A), ii) frequency (f) and iii) wavelength (λ).

y(x,t)=e^-(cx-dt) is the equation of a disturbance in a string where c and d are two positives and constant coefficients. Find the relation between c and d so that the equation y(x,t) satisfies the wave equation.

y(x,t)=(ax+bt)³ is the equation of a disturbance in a medium, where a and b are two positive and constant coefficients. Find the relation between a and b so that the equation y(x,t) satisfies the wave equation.

Consider a P-seismic wave traveling in the x direction and modeled by the equation D(x,t)=A cos (kx-ωt). Where the amplitude A is in meters, the wave vector k is in rad/m and the angular frequency ω in rad/s. Determine the axis along which the rock's particles oscillate.

A physics professor is demonstrating waves to his students. He takes a string, ties one end of it to a wall, and holds the other end in his hand. By shaking his hand, he creates a traveling wave on the string that can be described by W = 0.15 sin (4.2x - 28t). Here all the values are in SI units. Find the frequency of the wave.

A child is playing with a thread. She fixes one of the ends of the thread to a wall and holds the other end in her hand. She then moves her hand up and down which creates a traveling wave on the thread. If the wave is given by P = 0.15 sin (4.2x + 28t), calculate its velocity (both the magnitude and the direction). Note that all of the values are given in SI units.

One end of a string is tied to a tuning fork, and the other end is fixed to a wall. The tuning fork is struck, and it creates a traveling wave as shown in the figure. Given that the wave on the string is given by P (x,t) = (0.022 m) sin (32 m^-1 x - 1200 s^-1 t + 0.33), find the maximum acceleration of the string at x = 0.40 m.

A wave with a frequency of 500 Hz travels rightward through air with a speed of 340 m/s. Determine an equation for this wave if its amplitude is 0.03 cm, and D = -0.03 cm at t = 0, x = 0.

Some physics students are studying waves and vibrations. In the laboratory, they created a wave on a rope to observe its characteristics. The wave is described by the equation P (x, t) = 0.02 sin (40x - 1500t). Here, all the quantities are given in SI units. Find an expression for a wave having the same frequency, amplitude, and wavelength but that travels in the opposite direction.

In a physics lab, a wave generator produces a wave described by the equation D(x, t) = (0.15)sin(2.5x - 10.0t) where D represents the displacement of the string in meters, x is the distance along the string in meters, and t is the time in seconds. At a specific point x=0.50 m along the string and time t=0.30 seconds after the wave begins, determine the displacement, velocity, and acceleration of the string at that location.

A transverse wave is generated on a thin wire in a workshop experiment. The wave has a frequency of 230 Hz and a wavelength of 20.0 cm. What is the amplitude of the wave if it is observed that the maximum speed of particles on the wire is 0.12 times the wave speed?

A physics student is experimenting with a rope whose one end is fixed to a wall, and the other end is in his hand. The rope is in horizontal orientation. He generates a wave pulse on the rope by giving a slight shake. Afterward, the pulse travels along the rope to the right with a speed v = 1.5 m/s. At t = 0, the student notices that the shape of the pulse is described by the function P = 0.25 cos (2x + 1), where P and x are in meters. Find the formula for the wave pulse at any time t assuming there is no friction at play.

The figure below shows the water's surface in a tank with a sinusoidal wave at two different moments, t₁ = 0 and t₂ = 0.60 seconds. The wave is traveling to the right. Find the mathematical representation of this water wave. Assume that during this time interval, the wave's crest has moved 9.0 cm.

A violin's horizontal cord is held in place by two fixed ends: the bridge and the nut. The fundamental frequency of the stationary waves produced is 440 Hz. How long does it take the oscillating cord to go from the maximum positive displacement to the maximum negative displacement at a point located x = λ/4 from the left end?

A wire of length L is stretched between two clamps. The tension in the wire is adjusted until standing waves are observed on the wire. The amplitude and wavelength of the traveling waves are respectively  4 cm and 1.6 m. What is the amplitude of the wire particle located at i) 0.8 m, ii) 0.4 m, and iii) 0.2 m away from the left clamp?

A wire is stretched horizontally between two vertical stems. When it is strummed, the wire vibrates in its fundamental mode at a frequency of 95 Hz. The transverse waves on this stretched wire have a wavelength of 31 cm, a speed of 29.45 m/s, and an amplitude of 5 mm. What are the maximum transverse velocity and the maximum transverse acceleration of the wire particle at x = 15.5 cm, x = 7.75 cm, and x = 3.875 cm from the left stem?

A wire is strained at 145 N, and its mass per unit length is 0.008 kg/m. A sine wave sent on the wire has an amplitude of 4.5 cm and a wavelength of 1.3 m. Find the greatest speed of a mass element on the wire.

The figure below shows a segment of a wave traveling on a tensioned wire in the positive x-direction. The wave speed is 60 m/s. Find the velocity at positions a, b, and c at the instant the segment was captured.

When a sound wave travels through a wooden board, the positions of the wood particles at the instant t = 0 s are shown below. The equilibrium position between the wood particles is 4 cm. Sketch the wave displacement as a function of the position at t = 0 s.