Question

A long string carries a wave; a 6.00-m segment of the string contains four complete wavelengths and has a mass of 180g. The string vibrates sinusoidally with a frequency of 50.0 Hz and a peak-to-valley displacement of 15.0cm . (The "peak-to-valley" distance is the vertical distance from the farthest positive position to the farthest negative position.) (a) Write the function that describes this wave traveling in the positive x direction.

Answer 1

A long string carries the wave; a segment of the string measuring 6.00 meters in length and weighing 180 grams contains four full wavelengths. The string vibrates sinusoidally at a 50.0 Hz frequency with a 15.0 cm peak-to-valley displacement. The vertical distance between the farthest positive and negative positions is known as the "peak-to-valley" distance. The function that describes this wave traveling in the positive x direction is $y=(7.50 \mathrm{~cm}) \sin \left(\frac{4 \pi}{3} \times-314 t+\phi\right)$.

$\begin{aligned}& f=500 \mathrm{~Hz} \\&\mu=\frac{180 \times 10^{-3}}{6}=3 \times 10^{-2} K g / m \\&2 A=15 \times 10^{-2} m \\&A=7.5 \times 10^{-2} m \\&4 \lambda=6, \lambda=3 / 2 m \text {. and } v=f \lambda=75 m / \mathrm{sec} \\&\omega=2 \pi f=100 \pi \text { also } K=\frac{\omega}{v}=\frac{4 \pi}{3}\end{aligned}$

If phase constant is $\phi$ then

$\therefore$ equation is $y=7.5 \times 10^{-2} \sin \left[100 \pi t-\frac{4 \pi}{3} x+\phi\right]$

∴ equation is $y=(7.50 \mathrm{~cm}) \sin \left(\frac{4 \pi}{3} \times-314 t+\phi\right)$

• A wave is a disturbance that moves from one location to another, carrying energy but not always matter.
• The square of the equation provides a probability density map of the locations where an electron has a specific statistical likelihood to be at any given moment in time. Wave functions have no direct physical meaning.

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