The equation of a transverse wave traveling along a very...

The equation of a transverse wave traveling along a very long string is $y=6.0 \sin (0.020 \pi x+4.0 \pi t),$ where $x$ and $y$ are expressed in centimeters and $t$ is in seconds. Determine (a) the amplitude, (b) the wavelength, (c) the frequency, (d) the speed, (e) the direction of propagation of the wave, and $(\mathrm{f})$ the maximum transverse speed of a particle in the string. (g) What is the transverse displacement at $x=3.5 \mathrm{cm}$ when $t=$ 0.26 $\mathrm{s} ?$

The equation of a transverse wave traveling along a very long string is $y=6.0 \sin (0.020 \pi x+4.0 \pi t),$ where $x$ and $y$ are expressed in centimeters and $t$ is in seconds. Determine (a) the amplitude, (b) the wavelength, (c) the frequency, (d) the speed, (e) the direction of propagation of the wave, and $(\mathrm{f})$ the maximum transverse speed of a particle in the string. (g) What is the transverse
displacement at $x=3.5 \mathrm{cm}$ when $t=$
0.26 $\mathrm{s} ?$

Key Concepts

Wave Number and Angular Frequency

The wave number and angular frequency are parameters that describe how the wave varies in space and time, respectively. The wave number is associated with the spatial periodicity (wavelength) of the wave, while the angular frequency relates to the time periodicity (frequency) of the wave.

Transverse Particle Motion

In transverse waves, particles of the medium move perpendicular to the direction of wave propagation. The maximum transverse speed of a particle is obtained from the time derivative of the displacement, revealing how quickly points on the medium oscillate as the wave passes.

Wave Speed

Wave speed is the rate at which the disturbance or wave propagates through the medium. It can be determined by the relationship between wavelength and frequency, using the formula v = ?f. It is also related to the wave’s angular frequency and wave number.

Propagation Direction

The direction of wave propagation is inferred from the sign and terms in the phase of the wave equation. The way the phase expression is structured can indicate whether the wave travels in the positive or negative direction along the medium.

Wavelength

Wavelength is the spatial period of the wave—the distance over which the wave's shape repeats. It is the distance between consecutive points that are in phase, such as from crest to crest or trough to trough.

Amplitude

In wave mechanics, amplitude is the maximum displacement of a point on the wave from its equilibrium position. It represents the energy or strength of the wave and is a measure of how far the medium is displaced during the oscillation.

Frequency

Frequency is the number of oscillations or cycles that occur per unit time. It defines how many cycles pass a given point each second and is typically measured in hertz (Hz).