A wave is described by the equation y=(1 ·0 mm) sinπ((x)/(2 ·0 cm)-(t)/(0 ·01 s)) (a) Find th

step 1 High friends given wave equation is 1mm sign of by x y2 centimeter minus t upon 0 .01 second. Fi

A wave is described by the equation y=(1 ·0 mm) sinπ((x)/(2...

A wave is described by the equation $$ y=(1 \cdot 0 \mathrm{~mm}) \sin \pi\left(\frac{x}{2 \cdot 0 \mathrm{~cm}}-\frac{t}{0 \cdot 01 \mathrm{~s}}\right) $$ (a) Find the time period and the wavelength? (b) Write the equation for the velocity of the particles. Find the speed of the particle at $x=1 \cdot 0 \mathrm{~cm}$ at time $t=0 \cdot 01 \mathrm{~s}$. (c) What are the speeds of the particles at $x=3 \cdot 0 \mathrm{~cm}$, $5 \cdot 0 \mathrm{~cm}$ and $7 \cdot 0 \mathrm{~cm}$ at $t=0 \cdot 01 \mathrm{~s}$ ? (d) What are the speeds of the particles at $x=1 \cdot 0 \mathrm{~cm}$ at $t=0 \cdot 011,0 \cdot 012$, and $0 \cdot 013 \mathrm{~s} ?$

A wave is described by the equation
$$
y=(1 \cdot 0 \mathrm{~mm}) \sin \pi\left(\frac{x}{2 \cdot 0 \mathrm{~cm}}-\frac{t}{0 \cdot 01 \mathrm{~s}}\right)
$$
(a) Find the time period and the wavelength? (b) Write the equation for the velocity of the particles. Find the speed of the particle at $x=1 \cdot 0 \mathrm{~cm}$ at time $t=0 \cdot 01 \mathrm{~s}$.
(c) What are the speeds of the particles at $x=3 \cdot 0 \mathrm{~cm}$, $5 \cdot 0 \mathrm{~cm}$ and $7 \cdot 0 \mathrm{~cm}$ at $t=0 \cdot 01 \mathrm{~s}$ ?
(d) What are the speeds of the particles at $x=1 \cdot 0 \mathrm{~cm}$ at $t=0 \cdot 011,0 \cdot 012$, and $0 \cdot 013 \mathrm{~s} ?$

Key Concepts

Sinusoidal Wave Equation

A sinusoidal wave equation describes how a disturbance propagates through a medium using a sine (or cosine) function that depends on both space and time. This form of the equation encapsulates the periodic nature of the wave and is central to understanding wave phenomena in physics.

Angular Frequency and Wave Number

Angular frequency (?) and wave number (k) are key parameters in wave equations that describe the temporal and spatial oscillations of a wave, respectively. The angular frequency is related to the time period of the wave by T = 2?/?, while the wave number is connected to the wavelength by ? = 2?/k.

Time Period and Wavelength

The time period is the duration required for a point on the wave to complete one full oscillation, and the wavelength is the spatial period of the wave, representing the distance over which the wave's shape repeats. Both parameters are fundamental in quantifying the properties of periodic waves and are directly derived from the angular frequency and wave number.

Particle Velocity in Wave Motion

In a wave, the particles of the medium undergo oscillatory motion about their equilibrium positions. The velocity of these particles is determined by taking the partial derivative of the displacement function with respect to time, yielding the instantaneous rate of change of displacement at a fixed spatial point.

Differentiation in Wave Analysis

Differentiation, a basic tool from calculus, is crucial in wave analysis as it allows for the computation of quantities such as particle speed from displacement functions. By differentiating the wave equation with respect to time, one obtains the particle velocity, which is key to understanding the dynamics of wave motion.

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