15.8 A transverse harmonic wave on a string is described by

y(x, t) = 3.0 sin (36 t + 0.018 x + $π$ /4)

where x and y are in cm and t in s. The positive direction of x is from left to right.

(a) Is this a travelling wave or a stationary wave? If it is travelling, what are the speed and direction of its propagation?

(b) What are its amplitude and frequency?

(c) What is the initial phase at the origin?

(d) What is the least distance between two successive crests in the wave?

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Answered by

Vishal Baghel | Contributor-Level 10

5 months ago

The equation of a progressive wave travelling from right to left is given by the displacement function:

y (x, t) = a $\sin ? (ω t + k x + φ)$ ……….(i)

The given equation is

y(x, t) = 3.0 sin (36 t + 0.018 x + $π$ /4) …….(ii)

On comparing equations (i) and (ii), we find that the equation (ii) represents a travelling wave, propagating from right to left. Now, using equations (i) and (ii), we can write

$ω = 36 r a d / s$ , k = 0.018 $m^{- 1}$

From the relation $ν = \frac{ω}{2 π}$ and $λ = \frac{2 π}{k}$ and v = $ν λ$ , we can write

v = $\frac{ω}{2 π} \times$ $\frac{2 π}{k}$ = $\frac{ω}{k}$ = $\frac{36}{0.018}$ = 2000 cm /s = 20 m/s

Hence, the speed of the given travelling wave is 2

...more

The equation of a progressive wave travelling from right to left is given by the displacement function:

y (x, t) = a $\sin ? (ω t + k x + φ)$ ……….(i)

The given equation is

y(x, t) = 3.0 sin (36 t + 0.018 x + $π$ /4) …….(ii)

On comparing equations (i) and (ii), we find that the equation (ii) represents a travelling wave, propagating from right to left. Now, using equations (i) and (ii), we can write

$ω = 36 r a d / s$ , k = 0.018 $m^{- 1}$

From the relation $ν = \frac{ω}{2 π}$ and $λ = \frac{2 π}{k}$ and v = $ν λ$ , we can write

v = $\frac{ω}{2 π} \times$ $\frac{2 π}{k}$ = $\frac{ω}{k}$ = $\frac{36}{0.018}$ = 2000 cm /s = 20 m/s

Hence, the speed of the given travelling wave is 20 m/s

The frequency $ν = \frac{ω}{2 π}$ = $\frac{36}{2 π}$ = 5.73 Hz

Comparing equation (i) and (ii), we get

$φ$ = $\frac{π}{4}$

The distance between two successive crests or troughs is equal to the wavelength of the wave.

Wavelength $λ = \frac{2 π}{k}$ = $\frac{2 π}{0.018}$ = 349.07 cm = 3.49 m

less

The equation of a progressive wave travelling from right to left is given by the displacement function:y (x, t) = a <math><mrow><mrow><mi mathvariant="normal">sin</mi></mrow><mo>?</mo><mrow><mo>(</mo><mi>ω</mi></mrow></mrow><mi>t</mi><mo>+</mo><mi>k</mi><mi>x</mi><mo>+</mo><mi></mi><mi>φ</mi><mo>)</mo><mi></mi></math> ……….(i)The given equation isy(x, t) = 3.0 sin (36 t + 0.018 x + <math><mi>π</mi></math> /4) …….(ii)On comparing equations (i) and (ii), we find that the equation (ii) represents a travelling wave, propagating from right to left. Now, using equations (i) and (ii), we can write<math><mi>ω</mi><mo>=</mo><mn>36</mn><mi></mi><mi>r</mi><mi>a</mi><mi>d</mi><mo>/</mo><mi>s</mi></math> , k = 0.018 <math><msup><mrow><mrow><mi>m</mi></mrow></mrow><mrow><mrow><mo>-</mo><mn>1</mn></mrow></mrow></msup></math>From the relation <math><mi>ν</mi><mo>=</mo><mi></mi><mfrac><mrow><mrow><mi>ω</mi></mrow></mrow><mrow><mrow><mn>2</mn><mi>π</mi></mrow></mrow></mfrac></math> and <math><mi>λ</mi><mo>=</mo><mfrac><mrow><mrow><mn>2</mn><mi>π</mi></mrow></mrow><mrow><mrow><mi>k</mi></mrow></mrow></mfrac></math> and v = <math><mi>ν</mi><mi></mi><mi>λ</mi></math> , we can writev = <math><mfrac><mrow><mrow><mi>ω</mi></mrow></mrow><mrow><mrow><mn>2</mn><mi>π</mi></mrow></mrow></mfrac><mi></mi><mo>×</mo></math> <math><mfrac><mrow><mrow><mn>2</mn><mi>π</mi></mrow></mrow><mrow><mrow><mi>k</mi></mrow></mrow></mfrac></math> = <math><mfrac><mrow><mrow><mi>ω</mi></mrow></mrow><mrow><mrow><mi>k</mi></mrow></mrow></mfrac></math> = <math><mfrac><mrow><mrow><mn>36</mn></mrow></mrow><mrow><mrow><mn>0.018</mn></mrow></mrow></mfrac></math> = 2000 cm /s = 20 m/sHence, the speed of the given travelling wave is 20 m/sThe frequency <math><mi>ν</mi><mo>=</mo><mi></mi><mfrac><mrow><mrow><mi>ω</mi></mrow></mrow><mrow><mrow><mn>2</mn><mi>π</mi></mrow></mrow></mfrac></math> = <math><mfrac><mrow><mrow><mn>36</mn></mrow></mrow><mrow><mrow><mn>2</mn><mi>π</mi></mrow></mrow></mfrac></math> = 5.73 HzComparing equation (i) and (ii), we get<math><mi>φ</mi></math> = <math><mfrac><mrow><mrow><mi>π</mi></mrow></mrow><mrow><mrow><mn>4</mn></mrow></mrow></mfrac></math>The distance between two successive crests or troughs is equal to the wavelength of the wave.Wavelength <math><mi>λ</mi><mo>=</mo><mi></mi><mfrac><mrow><mrow><mn>2</mn><mi>π</mi></mrow></mrow><mrow><mrow><mi>k</mi></mrow></mrow></mfrac></math> = <math><mfrac><mrow><mrow><mn>2</mn><mi>π</mi></mrow></mrow><mrow><mrow><mn>0.018</mn></mrow></mrow></mfrac></math> = 349.07 cm = 3.49 m

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