A sinusoidal wave is described by the equation y1=(0.0800...

A sinusoidal wave is described by the equation $$y_{1}=(0.0800 \mathrm{~m}) \sin [2 \pi(0.100 x-80.0 t)]$$ where $y_{1}$ and $x$ are in meters and $t$ is in seconds. Write an expression for a wave that has the same frequency, amplitude, and wavelength as $y_{1}$ but which, when added to $y_{1}$, gives a resultant with an amplitude of $8 \sqrt{3} \mathrm{~cm}$.

A sinusoidal wave is described by the equation
$$y_{1}=(0.0800 \mathrm{~m}) \sin [2 \pi(0.100 x-80.0 t)]$$
where $y_{1}$ and $x$ are in meters and $t$ is in seconds. Write an expression for a wave that has the same frequency, amplitude, and wavelength as $y_{1}$ but which, when added to $y_{1}$, gives a resultant with an amplitude of $8 \sqrt{3} \mathrm{~cm}$.

Key Concepts

Sinusoidal Wave

A sinusoidal wave is a periodic oscillation that can be described using sine or cosine functions. It is characterized by parameters such as amplitude (the maximum displacement), frequency (how many oscillations occur per unit time), wavelength (the spatial period), and phase (the initial angle at t = 0). This fundamental concept is used to model various wave phenomena in physics.

Wave Superposition

The principle of superposition states that when two or more waves traverse the same space, their displacements add algebraically at every point. This principle is key in understanding how individual waves combine to form a resultant wave, leading to phenomena such as interference patterns.

Phase Shift

A phase shift refers to the horizontal translation of a sinusoidal function along the time or space axis. Changing the phase effectively shifts the position of the wave, which can alter the interference outcome when two waves of the same frequency and amplitude overlap.

Interference of Waves

Interference occurs when two or more waves overlap and combine, resulting in a new wave pattern. Depending on the relative phase difference between the waves, the interference can be constructive, where the amplitudes add to produce a larger resultant amplitude, or destructive, where they cancel each other out partly or completely.

Trigonometric Addition for Waves

When combining sinusoidal waves, trigonometric identities are used to simplify the expression for the sum of the waves. In particular, identities that relate the sum of sine functions to a product involving cosine provide insight into how phase differences affect the total amplitude of the superposed wave.

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