The equation of a transverse wave traveling along a very long string is y=9.24 sin(0.0317x+ 3.48t), where x and y are expressed in centimeters and t is in seconds. Determine (a) the amplitude, (b) the wavelength, (c) the frequency, (d) the speed, (e) the direction of propagation of the wave and (f) the maximum transverse speed of a particle in the string. (g) What is the transverse displacement at x 5.98 cm when t = 0.792 s?
Properties of sound
A sound wave is a mechanical wave (or mechanical vibration) that transit through media such as gas (air), liquid (water), and solid (wood).
Quality Of Sound
A sound or a sound wave is defined as the energy produced due to the vibrations of particles in a medium. When any medium produces a disturbance or vibrations, it causes a movement in the air particles which produces sound waves. Molecules in the air vibrate about a certain average position and create compressions and rarefactions. This is called pitch which is defined as the frequency of sound. The frequency is defined as the number of oscillations in pressure per second.
Categories of Sound Wave
People perceive sound in different ways, like a medico student takes sound as vibration produced by objects reaching the human eardrum. A physicist perceives sound as vibration produced by an object, which produces disturbances in nearby air molecules that travel further. Both of them describe it as vibration generated by an object, the difference is one talks about how it is received and other deals with how it travels and propagates across various mediums.
Please show all the steps and equations you use and their units.
![### Transverse Wave Analysis
The equation of a transverse wave traveling along a very long string is given by:
\[ y = 9.24 \sin(0.0317\pi x + 3.48\pi t) \]
where \( x \) and \( y \) are expressed in centimeters (cm) and \( t \) is in seconds (s). We need to determine the following parameters:
#### (a) Amplitude
**Given**: 9.24 cm
The amplitude of the wave is the maximum displacement from the equilibrium position, which here is 9.24 cm.
#### (b) Wavelength
**Calculated Value**: 63.0915 cm
The wavelength (\(\lambda\)) can be determined from the wave number \(k\).
\[ k = 0.0317\pi \]
\[ \lambda = \frac{2\pi}{k} = \frac{2\pi}{0.0317\pi} \approx 63.0915 \text{ cm} \]
#### (c) Frequency
**Calculated Value**: 1.74 rad/s or s\(^{-1}\)
The angular frequency (\(\omega\)) is given by:
\[ \omega = 3.48\pi \]
The frequency (\(f\)) is:
\[ f = \frac{\omega}{2\pi} = \frac{3.48\pi}{2\pi} = 1.74 \text{ rad/s or s}^{-1} \]
#### (d) Speed
**Calculated Value**: 109.779 cm/s
The speed (\(v\)) of the wave can be calculated using:
\[ v = \lambda \cdot f \]
Given:
\[ \lambda = 63.0915 \text{ cm} \]
\[ f = 1.74 \text{ s}^{-1} \]
\[ v = 63.0915 \text{ cm} \times 1.74 \text{ s}^{-1} \approx 109.779 \text{ cm/s} \]
#### (e) Direction of Propagation
**Given**: -x direction
The direction of propagation is determined by the sign in front of the wave number term in the equation. Here, it moves in the -x direction.
#### (f) Maximum Transverse Speed of a Particle in the String](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4f3fcee3-df4f-43f5-a70c-56c5ba155c30%2Fbc853383-5a69-440a-9870-602c98173b07%2Fi636brv_processed.png&w=3840&q=75)
Step by step
Solved in 3 steps with 2 images
This answer is incomplete. I need help with f and g.
by Bartleby Expert
