Wave Types: Mechanical, Electromagnetic, and Matter

• A wave is a disturbance that travels or propagates.

Types of Waves

• Mechanical waves require a medium for propagation and cannot propagate through a vacuum e.g. waves on a string, water waves, and sound waves.
• Electromagnetic waves do not necessarily require a medium and can travel through a vacuum e.g. light, radio waves, and X-rays, and have a speed C = 3 x 10^8 m/s in a vacuum.
• Matter waves are associated with matter in motion of electrons, protons, neutrons, atoms, and molecules.

Mechanical Waves

• Motion of a mechanical wave involves oscillations of constituents of the medium.
• Transverse waves occur when the constituents of the medium oscillate perpendicularly to wave propagation direction e.g. a pulse on a stretched string.
• Longitudinal waves occur when the constituents of the medium oscillate along the direction of wave propagation e.g. sound waves where compressions and rarefactions occur.

Speed of Waves

• Speed of a mechanical wave is determined by the inertial (mass density) and elastic properties (Young's, Bulk, and Shear modulus) of the medium.
• Speed of a transverse wave on a stretched string is determined by the restoring force (Tension) set up in the medium (string) when it is disturbed and its inertial property (linear mass density).
• The formula for speed is given by v = sqrt(T/μ), where T is tension in the string and μ is mass per unit length.
• Dimensional analysis gives [LT^-1]

Speed of Longitudinal Waves

• Longitudinal waves (sound) travel in the form of compressions and rarefactions of small volume elements of air.
• The elastic property determining stress under compressional strain is the Bulk modulus defined as B = -ΔP/(ΔV/V)
• Where ΔP = a change in pressure and ΔV proportional to the change in volume
• The speed of sound 'v' equals sqrt(B/ρ) where B is the Bulk modulus and ρ the mass density; this formula is called Newton's formula.
• The value obtained using this formula is about 15% less than the experimental value of 331m/s.
• Laplace suggested pressure variations in sound waves propagation are so fast that there is very little time for heat flow to maintain constant temperature, and the variations are adiabatic, not isothermal.

Adiabatic Process

• For an adiabatic process, PV^γ = constant, where γ is the adiabatic index.
• The speed of sound in a gas can ben be written as v = sqrt(γP/ρ)
• In air, γ approximately equals 1.4.

Plane Progressive Wave

• The equation of a plane progressive wave describes a sinusoidal displacement relation.
• The equation is to describe a sinusoidal travelling wave as a function of both position x and time t.
• x is the position of the constituents of the medium.
• y is the displacement from the equilibrium position.
• The equation for the displacement is: y(x,t) = a sin(kx - ωt + φ). φ = phase constant, k = angular wave number, ω = angular frequency
• The shape of the wave at any instant is a sine wave
• At a fixed location, displacement varies sinusoidally
• As t increases, x must increase to keep kx - ωt + φ constant.

Wave Properties

• Amplitude is the maximum displacement of constituents of the medium from their equilibrium position.
• The quantity (kx - ωt + φ) determines displacement of the wave at any position ‘x’ and at any instant ‘t’ is called phase.
• Wavelength (λ) is the minimum distance between two points having the same phase.
• The displacement y for a point 'x' at t = 0 is given by y(x, 0) = a sin kx.
• Since sine function repeats its value after every 2π change in angle a sin kx = a sin (kx + 2π)

Wavelength and Angular Wave Number

• Displacement remains same at x and x + λ i.e. a sin kx = a sin k(x + λ) where x + λ = 2π/k are the same angular displacement
• The Wave number, 'k' = 2π/λ.
• Radians per meter are the units for the wave number.
• Period 'T' is s the time for one complete oscillation.
• Angular frequency is ω = 2π/T and has SI units 'rad/s.'
• Frequency indicates ‘no. of oscillations per second.’ Frequency, v = 1/T (unit s^-1 or Hertz).

Traveling Wave Speed

• The speed gives the shift to the entire wave pattern
• The speed of the wave = velocity 'v' = Δx/Δt = ω/k = 2π/T / 2π/λ velocity 'v' = λ/T = vλ

Superposition Principle

• Principle states that, whenever two or more waves meet at a point (x,t) their resultant displacement is the algebraic sum due to each wave.

Interference of Waves

• Consider two progressive waves y1 and y2 travelling along +x axis, having same ω, k and amplitude but of different initial phase.
• Given the condition that y1(x,t) = a sin (kx-ωt) and y2(x,t) = a sin (kx-ωt + φ)
• The amplitude A = 2a cos(φ/2); A(φ) is the dependence of the resultant amplitude on the phase difference φ between the two interfering waves.
• Thus resultant is also a wave of same ω & k, but different initial phase, which means it is travelling along + x axis.
• From those conditions, two cases arise: Constructive interference and destructive interference.

Constructive Interference

• It arises when |cos(φ/2)| reaches its maximun value of 1; when φ = 0, 2π....
• For φ = 0; A(φ) = 2a = max
• Waves are in in phase.

Destructive interference

• Destructive interference is the opposite case to constructive interference arising when |cos(φ/2)| is the minimum which equals 0.
• φ/2 equals to π/2, 3π/2 or φ = π, 3π, 5π....
• For φ = π; A(φ) = 0 i.e the waves are out of phase (by 180 degrees).

Reflection of Waves

• When a progressive wave meets a rigid (fixed) boundary, the reflected wave has the same shape but it suffers a phase change of π or 180° upon reflection.
• The net maximum displacement and the rigid is twice the amplitude.

Standing Waves, Fixed Boundary

• If the incident wave is, yi(x,t) = a sin(kx-ωt)
• At a rigid boundary, the reflected wave is given by yr(x,t) = a sin (kx+ωt + π) = - a sin (kx+ωt).
• Boundary Not Rigid: The boundary is not rigid but completely free to move.
• as in the case of a string tied to a freely moving ring on a rod.
• It then has the same phase and amplitude as the incident wave

Wave Speed For Standing Waves

• Standing waves are described by when waves travelling in opposite directions, superimpose on each other, they produce a steady (constant) wave pattern called standing or stationary waves.
• The waves can be represented as L→R y1(x,t) = a sin (kx-ωt), R→L y2(x,t) = a sin (kx+ωt) with the resultant wave being yr(x,t) = y1(x,t) + y2(x,t).
• Which after using trigonometric identities becomes: a [ sin(kx-ωt) + sin(kx+ωt)]
• The term is now dependent on both time and location.
• the combination indicates that the wave is both progressive and travelling
• This term has kx and ωt separately, this means that : The resultant is not a progressive /traveling wave and is called a Standing or Stationary wave, but all points in the medium oscillate with the same w (or T).
• Amplitude is fixed at each position.

Standing Waves: Nodes

• Nodes are the positions at which amplitude is always zero.
• The node can be calculated by: A(x) = 2a sin kx == 0 (where A(x): indicates Amplitude as it depends on the amplitude)
• Hence Kx = nπ (where “ n = 0, 1, 2,3…”)

Standing Waves:Antinodes

• Antinodes are the points where the amplitudes are maximum.
• The condition for this scenario arises when: amplitudes largest A(n2x).

Standing Waves: Strings

• Stretched String: Consider a stretched string of definite length fixed at both ends.

Standing Waves: Open Pipes Conditions

• Standing waves can be generated in a pipe with both ands open.

• For those condition one of the conditions for either and wave must have x=0 which corresponds to 2a sin, and satisfied while at XL, A(x) =0 when sin=0.

• So, A equals ‘na/2 i.e. x=n' (n21, 23.

• In conclusion: The formula V equals, (nn" where A as a fixed value depends can vary. This formula represents all corresponding frequency in the relation.

Standing Waves: Closed Pipes Conditions

• Standing waves can be generated in a pipe with one end closed and the other open.
• The pressure changes at the closed end which is greater at one end than the end that is at zero displacement.
• in addition there is an amplitude and it satisfies an anti-node relation.

Beats

• Beats occur when two harmonic sound waves of close (but not equal) frequencies travel in the same direction
• We hear a sound of similar frequency, but varying intensity.

The superposition of waves

• The formula states The superposition of waves may follow the principle S = s,+s, =asin2xt which is then averaged. By the principle, this leads to sin( A + Sinb = 2 cos CD

Explore the different types of waves, including mechanical, electromagnetic, and matter waves. Understand the characteristics of transverse and longitudinal waves, with examples. Learn how the speed of a mechanical wave is determined.