Question

A harmonic wave on a string is described by

Answer 1

$\huge{\bold{\orange{\underline{ Solution }}}}$

Given :-

A harmonic wave on a string is described by

$\sf{ Y( x, t) = 0.1 sin(300t + 0.01x + π/3)}$

• x is in cm and t is in seconds

Answer 1 :-

Equation for travelling wave :-

$\sf{ Y( x, t) = Asin(ωt + kx + Φ)...eq(1)}$

Equation for stationary wave :-

$\sf{ Y( x, t) = Acos(ωt - kx )...eq(2)}$

Given equation for wave :-

$\sf{ Y( x, t) = 0.1 \:sin(300t + 0.01x + π/3)...eq(3)}$

On comparing eq(1) , (2) and (3)

We can conclude that, Given wave represent travelling wave.

Answer 2 :-

From solution 1 , We can say that,

$\sf{ Y( x, t) = 0.1 \: sin(300t + 0.01x + π/3).}$

It is travelling from right to left direction

Hence, The direction of its propagation is right to left that is towards +x direction.

Answer 3 :-

Here, We have to find the wave period

We know that,

Wave period = wavelength / velocity

Wave equation :-

$\sf{ Y( x, t) = 0.1 \:sin(300t + 0.01x + π/3).}$

• ω = 300rad/s
• k = 0.01

We know that,

$\sf{v =}{\sf{\dfrac{ ω}{2π}}}{\sf{\: and\:}}{\sf{ λ =}}{\sf{\dfrac{ 2π}{k}}}$

Subsitute the required values,

$\sf{ wave\: period =}{\sf{\dfrac{ 2π/k}{ω/2π }}}$

$\sf{ wave \:period = }{\sf{\dfrac{k}{ω}}}$

$\sf{ wave\: period =}{\sf{\dfrac{ 0.01}{300}}}$

$\sf{ wave\: period = 0.000033\: s}$

Answer 4 :-

The wavelength of given wave

$\bold{ λ = }{\bold{\dfrac{2π}{k}}}$

Subsitute the required values,

$\sf{ λ = }{\sf{\dfrac{2 × 3.14 }{0.01}}}$

$\sf{ λ = }{\sf{\dfrac{6.28}{0.01}}}$

$\sf{ λ = 628 \: cm }$

Answer 5 :-

We have wave equation

$\sf{ Y( x, t) = 0.1 sin(300t + 0.01x + π/3).}$

Travelling wave equation :-

$\sf{ Y( x, t) = A\:sin(ωt + kx + Φ)...eq(1)}$

Therefore,

Amplitude of the wave particle

$\sf{ A = 0.1 \: cm}$

Hence, The amplitude of the particle is 0.1 cm