The function y(x, t)=(15.0 cm) cos(πx-15 πt), with x in meters and t in seconds, describes a wav

step 1 16 .12. So we're given the function for a wave on a string, x and t are in meters and seconds re

The function y(x, t)=(15.0 cm) cos(πx-15 πt), with x in...

The function $$y(x, t)=(15.0 \mathrm{cm}) \cos (\pi x-15 \pi t),$$ with $x$ in meters and $t$ in seconds, describes a wave on a taut string. What is the transverse speed for a point on the string at an instant when
that point has the displacement $y=+12.0 \mathrm{cm} ?$

Key Concepts

Wave Equation

A wave equation describes how a disturbance propagates through a medium. In the context of a taut string, the displacement of the string is modeled using a sinusoidal function of both space and time. This general framework helps us analyze wave behavior, including amplitude, wavelength, and frequency, which are core concepts in wave mechanics.

Transverse Velocity

Transverse velocity refers to the speed at which a point on the medium moves perpendicular to the direction of wave propagation. It is obtained by differentiating the wave displacement with respect to time. Understanding transverse velocity is key to analyzing energy propagation and local dynamics of a medium under wave motion.

Differentiation of Wave Functions Using the Chain Rule

Calculating the transverse speed of a point on the string involves taking the time derivative of the wave function. This requires the use of the chain rule, as the wave function is a composite function of time and space. Mastering this technique is essential for determining not only velocities but also accelerations in systems governed by periodic functions.

Phase Relationships in Harmonic Waves

The phase of a harmonic wave, which combines the spatial and temporal components, plays a crucial role in determining the instantaneous displacement and velocity of a point on the string. By setting the displacement equal to a specified value, one can solve for the phase and subsequently evaluate the relevant dynamic properties, such as transverse speed.

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