(II) The pressure variation in a sound wave is given by ΔP=0.0035 sin(0.38 πx-1350 πt) where ΔP

step 1 Hello and welcome to this video solution of numerate. Here it's given that the pressure variatio

(II) The pressure variation in a sound wave is given by...

(II) The pressure variation in a sound wave is given by $\Delta P=0.0035 \sin (0.38 \pi x-1350 \pi t)$ where $\Delta P$ is in pascals, $x$ in meters, and $t$ in seconds. Determine $(a)$ the wavelength, $(b)$ the frequency, $(c)$ the speed, and $(d)$ the displacement amplitude of the wave. Assume the density of the medium to be $\rho=2.3 \times 10^{3} \mathrm{kg} / \mathrm{m}^{3} .$

(II) The pressure variation in a sound wave is given by
$\Delta P=0.0035 \sin (0.38 \pi x-1350 \pi t)$
where $\Delta P$ is in pascals, $x$ in meters, and $t$ in seconds. Determine $(a)$ the wavelength, $(b)$ the frequency, $(c)$ the speed, and $(d)$ the displacement amplitude of the wave. Assume the density of the medium to be $\rho=2.3 \times 10^{3} \mathrm{kg} / \mathrm{m}^{3} .$

Key Concepts

Wave Speed

Wave speed is the rate at which the wave propagates through a medium. It is determined by multiplying the wavelength by the frequency (or equivalently dividing the angular frequency by the wavenumber). In the context of sound and other mechanical waves, the medium’s properties, such as density and elasticity, largely influence the wave speed.

Sinusoidal Wave Equation

A sinusoidal wave equation represents a periodic disturbance that varies with space and time using sine (or cosine) functions. It typically includes parameters like amplitude, wave number, angular frequency, and phase that describe the physical characteristics of the wave. This form is fundamental in understanding the propagation of waves in various media.

Wavelength

Wavelength is the distance between two consecutive points in phase on a wave (such as crests or troughs). It is directly related to the spatial frequency of the wave, with the wavenumber k given by 2? divided by the wavelength. Wavelength is a key parameter in describing the spatial properties of wave propagation.

Frequency

Frequency defines the number of oscillations or cycles a wave completes per unit time, measured in hertz (Hz). It is related to the angular frequency through the relationship ? = 2?f. Understanding frequency is essential for characterizing how fast a wave oscillates and for linking time-dependent aspects of wave behavior.

Acoustic Displacement Amplitude

Acoustic displacement amplitude quantifies the maximum displacement of particles in a medium due to the passage of a sound wave. It is related to the pressure amplitude by a formula that involves the medium’s density, the speed of the wave, and the angular frequency. This concept bridges the pressure variations observed in a sound wave with the actual movement of the medium's particles.

Transcript

Please give Ace some feedback