Question

plane progressive wave definition with derivatives


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Answer 1

Answer:

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Progressive Wave

A progressive wave is defined as the onward transmission of the vibratory motion of a body in an elastic medium from one particle to the successive particle.

Wave

Equation of a plane progressive wave

An equation can be formed to represent generally the displacement of a vibrating particle in a medium through which a wave passes. Thus each particle of a progressive wave executes simple harmonic motion of the same period and amplitude differing in phase from each other.

Let us assume that a progressive wave travels from the origin O along the positive direction of X axis, from left to right (Fig. 7.6). The displacement of a particle at a given instant is

y = a sin ωt …... (1)

where a is the amplitude of the vibration of the particle and ω = 2πn.

The displacement of the particle P at a distance x from O at a given instant is given by,

y = a sin (ωt - φ) …... (2)

If the two particles are separated by a distance λ, they will differ by a phase of 2π. Therefore, the phase φ of the particle P at a distance

x is φ = (2π/λ) x

y = a sin (ωt - 2πx/λ) …... (3)

Since ω = 2πn = 2π (v/λ), the equation is given by,

y = a sin [(2πvt/λ) - (2πx/λ)]

y = a sin 2π/λ (vt – x) …... (4)

Since, ω = 2π/T, the equation (3) can also be written as,

y = a sin 2π (t/T – x/λ) …... (5)

If the wave travels in opposite direction, the equation becomes,

y = a sin 2π (t/T + x/λ) …... (6)

Explanation:

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Answer 2

$\huge\mathfrak {ANSWER:-}$

Plane progressive wave is the disturbance produced in the medium travels onward, it being handed over from one particle to the next.

$\huge\underline{Derivatives:-}$

Equation of a plane progessive wave

An equation can be formed to represent generally the displacement of a vibrating particle in a medium through which a wave passes. Thus each particle of a progressive wave executes simple harmonic motion of the same period and amplitude differing in phase from each other.

Let us assume that a progressive wave travels from the origin O along the positive direction of X axis, from left to right (Fig. 7.6). The displacement of a particle at a given instant is

y = a sin ωt …... (1)

where a is the amplitude of the vibration of the particle and ω = 2πn.

The displacement of the particle P at a distance x from O at a given instant is given by,

y = a sin (ωt - φ) …... (2)

If the two particles are separated by a distance λ, they will differ by a phase of 2π. Therefore, the phase φ of the particle P at a distance

x is φ = (2π/λ) x

y = a sin (ωt - 2πx/λ) …... (3)

Since ω = 2πn = 2π (v/λ), the equation is given by,

y = a sin [(2πvt/λ) - (2πx/λ)]

y = a sin 2π/λ (vt – x) …... (4)

Since, ω = 2π/T, the equation (3) can also be written as,

y = a sin 2π (t/T – x/λ) …... (5)

If the wave travels in opposite direction, the equation becomes,

y = a sin 2π (t/T + x/λ) …... (6)