problem 4: Show that the product of force and velocity has the same units of power. Finish the calculation below. Sound intensity is (time averaged) T 1 I(x) = T P(t)v(t)dt T -/ povo? A² cos² (kx – wt + po)dt
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.
![**Problem 4:** Show that the product of force and velocity has the same units of power. Finish the calculation below.
Sound intensity is (time averaged):
\[ I(x) = \frac{1}{T} \int_{0}^{T} P(t)v(t) \, dt \]
\[ = \frac{1}{T} \int_{0}^{T} \rho_0 v \omega^2 A^2 \cos^2 (kx - \omega t + \phi_0) \, dt \]
Here, detailed explanations of the parameters and steps for further calculations should be provided for clarity:
- \(I(x)\) represents the sound intensity at position \(x\).
- \(P(t)\) is the instantaneous pressure at time \(t\), and \(v(t)\) is the instantaneous velocity at time \(t\).
- \(\rho_0\) is the mean density of the medium through which the sound wave is propagating.
- \(v\) is the speed of sound in the medium.
- \(\omega\) is the angular frequency of the sound wave.
- \(A\) is the amplitude of the particle displacement.
- \(\cos^2\) function represents the square of the cosine of the phase term, where \(kx\) is the wave number, \(\omega t\) is the angular frequency multiplied by time, and \(\phi_0\) is the phase angle.
The calculation further simplifies the expression to find the average sound intensity over one period \(T\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa49e8544-22cd-49e6-b20c-ed062c173676%2Fa0784340-9d0b-4d64-93d2-6871a1f7694d%2F7lteydh_processed.png&w=3840&q=75)
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