A sinusoidal transverse wave traveling in the positive...

A sinusoidal transverse wave traveling in the positive direction of an $x$ axis has an amplitude of $2.0 \mathrm{cm},$ a wavelength of $10 \mathrm{cm},$ and a frequency of 400 $\mathrm{Hz}$ . If the wave equation is of the form $y(x, t)=y_{m} \sin (k x \pm \omega t),$ what are (a) $y_{m},$ (b) $k,$ (c) $\omega,$ and (d) the correct choice of sign in front of $\omega ?$ What are (e) the maximum transverse speed of a point on the cord and (f) the speed of the wave?

A sinusoidal transverse wave traveling in the positive direction of an $x$ axis has an amplitude of $2.0 \mathrm{cm},$ a wavelength of
$10 \mathrm{cm},$ and a frequency of 400 $\mathrm{Hz}$ . If the wave equation is of the form $y(x, t)=y_{m} \sin (k x \pm \omega t),$ what are (a) $y_{m},$ (b) $k,$ (c) $\omega,$ and
(d) the correct choice of sign in front of $\omega ?$ What are (e) the maximum transverse speed of a point on the cord and (f) the speed of
the wave?

Key Concepts

Wave Speed

The wave speed is the speed at which the wave propagates along the medium. It is determined by the relationship between the angular frequency and the wave number, expressed as v = ?/k. This speed indicates how fast the overall shape of the wave moves through space.

Maximum Transverse Speed

The maximum transverse speed is the highest speed at which a point on the medium oscillates perpendicular to the direction of wave travel. It is obtained by differentiating the wave equation with respect to time and is given by the product of the angular frequency and the amplitude (?y_m).

Phase and Sign Convention

The phase component (kx ± ?t) in the wave equation determines the position and the timing of the peaks and troughs. The correct sign in front of the ?t term is chosen based on the wave's direction of propagation: a negative sign indicates movement in the positive x direction, ensuring that points of constant phase satisfy x - (?/k)t = constant.

Amplitude

Amplitude is the maximum displacement from the equilibrium position that the particles in the medium achieve due to the wave. It is a measure of the energy carried by the wave and appears as the coefficient in the sinusoidal wave equation.

Wave Number

The wave number, denoted as k, represents the spatial frequency of the wave. It is defined as 2? divided by the wavelength (k = 2?/?) and indicates the number of radians of phase change per unit distance along the wave.

Angular Frequency

Angular frequency, denoted as ?, quantifies the temporal aspect of the wave. It is defined as 2? times the frequency (? = 2?f), representing the rate at which the wave oscillates in time in radians per second.

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