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

The equation of a transverse wave traveling along a string is $$y=0.15 \sin (0.79 x-13 t)$$ in which $x$ and $y$ are in meters and $t$ is in seconds. (a) What is the displacement $y$ at $x=2.3 \mathrm{m}, t=0.16 \mathrm{s} ?$ A second wave is to be added to the first wave to produce standing waves on the string. If the second wave is of the form $y(x, t)=y_{m} \sin (k x \pm \omega t),$ what are (b) $y_{m}$ (c) $k,(\mathrm{d}) \omega,$ and $(\mathrm{e})$ the correct choice of sign in front of $\omega$ for this second wave? (f) What is the displacement of the resultant standing wave at $x=2.3 \mathrm{m}, t=0.16 \mathrm{s} ?$

The equation of a transverse wave traveling along a string is
$$y=0.15 \sin (0.79 x-13 t)$$ in which $x$ and $y$ are in meters and $t$ is in seconds. (a) What is the displacement $y$ at $x=2.3 \mathrm{m}, t=0.16 \mathrm{s} ?$ A second wave is to be added
to the first wave to produce standing waves on the string. If the second wave is of the form $y(x, t)=y_{m} \sin (k x \pm \omega t),$ what are (b) $y_{m}$ (c) $k,(\mathrm{d}) \omega,$ and $(\mathrm{e})$ the correct choice of sign in front of $\omega$ for this second wave? (f) What is the displacement of the resultant standing
wave at $x=2.3 \mathrm{m}, t=0.16 \mathrm{s} ?$

Key Concepts

Wave Propagation Direction

The sign in front of the time-dependent angular frequency term in the wave equation indicates the direction of wave propagation. A negative sign in front of ?t typically represents a wave traveling in the positive x-direction, whereas a positive sign indicates travel in the opposite direction. When setting up conditions for standing wave formation, correctly choosing the sign is crucial to ensure the new wave travels in the opposing direction to the original wave.

Phase and Displacement Evaluation

The phase of a wave, expressed as the argument of the sine (or cosine) function, includes contributions from spatial and time variables. Evaluating the displacement at specific positions and times involves substituting the x and t values into the phase. This process is crucial for predicting the instantaneous displacement of the medium at any given point and time.

Standing Waves Formation

Standing waves are formed by the superposition of two waves with identical amplitudes, frequencies, and wavelengths traveling in opposite directions. The interference between the forward and backward moving waves leads to a pattern of nodes, where displacement is always zero, and antinodes, where the displacement reaches a maximum. This phenomenon is essential in understanding resonance and vibration in confined media.

Transverse Waves

Transverse waves are disturbances that propagate through a medium in which the displacement of the medium is perpendicular to the direction of wave travel. In the given equation, the string's displacement is described by a sine function, indicating that every point on the string oscillates perpendicular to the length of the string as the wave travels.

Wave Equation Parameters

The wave equation is characterized by parameters such as amplitude, wave number, and angular frequency. The amplitude determines the maximum displacement from equilibrium, the wave number (k) relates to the wavelength via k = 2?/?, and the angular frequency (?) is related to the period by ? = 2?/T. These parameters collectively define the shape and speed of the wave.

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