Assam Don Bosco University Fundamentals of Wave Motion Parag Bhattacharya Department of Basic Sciences School of Engineering and Technology

The Travelling Wave

Suppose we consider any disturbance moving along 1D using the symbol Ψ as follows The profile or shape of the same wave at t = 0 can be represented as:

Consider a pulse of any arbitrary shape, moving with a speed v along the positive x- axis in some reference frame S. Now, consider another reference frame S' that is moving along with the pulse, also with a speed v. In the frame S', the pulse is stationary. Hence, In time t, the pulse as well as frame S' have moved a distance vt along x. And, x' = x – vt Therefore,

Therefore, a progressive wave-function is represented by functions of x and t which are of the form: subject to the condition that ψ is finite everywhere and at all times. To obtain a 1D wave-function: Step 1: Choose the desired profile function f(x) Step 2: Substitute x with a) x – vt (for a wave moving along the positive x- axis) b) x + vt (for a wave moving along the negative x- axis) In general, a wave function takes the form:

+ x– x ψ + x– x ψ = f(x) = f(x – vt)

+ x– x ψ + x– x ψ = f(x) = f(x + vt)

Example 1Example 2 Given a profile function The corresponding wave functions are: For a wave travelling along the positive x-axis For a wave travelling along the negative x-axis For a wave travelling along the positive x-axis For a wave travelling along the negative x-axis

The Differential Wave Function

Consider the travelling disturbance as where Differentiating (1) with respect to x...(1)...(2) Again, differentiating (1) with respect to t...(3) From (2) and (3), we get,...(4)

Differentiating (2) wrt x or x'...(5) Differentiating (3) wrt t...(6) Using (3) in (6)...(7)

Finally, using (5) in (7), we obtain the 2nd order differential wave equation for a 1D travelling wave moving in an undamped system or

Travelling Harmonic Waves

In a harmonic wave, the disturbance at any point is harmonic in nature i.e., follows a sinusoidal variation. Hence we choose a sinusoid as the profile function: where k and A are positive constants (k is known as the angular wave number, and A is known as the amplitude) The resulting travelling wave function is:...(1) Since a harmonic wave is periodic with respect to space and time, we define: Wavelength λ:total length per wave Time/temporal period τ:total time per wave

ψ = A sin(kx) + x – x 0 ψ = A sin(kx) + x – x 0 = A sin[k(x – vt)]

ψ = A sin(kx) + x – x 0 ψ + x – x 0 = A sin[k(x + vt)]

Thus, Therefore, Which implies, Because the wave-function ψ is periodic with respect to position

Therefore, Which implies, Because the wave-function ψ is periodic with respect to time

Basic definitions: Temporal frequency: Temporal angular frequency: Spatial frequency: Spatial angular frequency:

Various representations of the same harmonic travelling wave:

The most general form of the travelling harmonic wave is: Here, the phase of the wave is: And the initial phase, i.e., the phase at x = 0 and t = 0 is:

The phase of the wave is: The rate of change of phase with time (keeping x constant): The rate of change of phase with position (keeping t constant): If A(x, y), then If phase φ is constant, then Thus, the phase velocity is

The Supersposition Principle If ψ 1 and ψ 2 are two separate solutions to the differential wave equation, then any linear combination of ψ 1 and ψ 2 is also a solution, i.e., c 1 ψ 1 + c 2 ψ 2 also satisfies the differential equation. Since ψ 1 and ψ 2 are individual solutions, thus, and...(1)...(2) Adding (1) and (2), we obtain,

Representation of waves using complex numbers

A complex number has the form: where Real part: Imaginary part: 0 Im(z) Re(z) z r Cos θ r Sin θ θ In terms of polar coordinates Thus,

If r = 1, Therefore, Thus, This is known as the Euler's formula Summary:

Given a complex number: Its complex conjugate is: From Euler's formula, we have, Thus, its complex conjugate is...(1)...(2) Adding (1) and (2) yieldsSubtracting (2) from (1) yields

Taking this further: The modulus of a complex number is and

Evaluate: 1) 2) 3) 4) 5) 6)

Thus, e z is periodic, repeats every i2π

A wave can be represented as the real part of a complex harmonic function: Or as the imaginary part: However, for computational ease, we represent the wave as: It is easier to perform all calculations using complex exponentials. After arriving at the final result, to represent the actual wave, we extract either the real part or the imaginary part.

Slides available at: pararover.wordpress.com/2015/08/07/ fundamentals-of-wave-motion/

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