Physics

Wave phenomena

Study of waves, characteristic parameters, wave equation, double periodicity, energy and intensity of waves.

  • Wave: is the propagation of a disturbance through space with transport of energy but not of matter.
  • Mechanical waves: require a medium to transmit.
  • Electromagnetic waves: do not require a medium.
  • Transverse wave: the direction of oscillation is transverse to the direction of propagation. Ex.: vibration of a string.
  • Longitudinal wave: the direction of oscillation is the same as the direction of propagation. Ex.: sound.
  • Displacement (x or y): is the position of the particle with respect to the equilibrium position (with respect to the center of vibration). It is measured in m in the SI, and is represented by the letter x if the wave is longitudinal, or by the letter y if the wave is transverse. Displacement is a function of time at each point in space x(x, t) or y(x, t).
  • Amplitude (A): is the maximum displacement, that is, the maximum separation from the equilibrium point of the vibrating particles. We represent it by the letter A and it is also measured in m.
  • Period (T): is the time it takes the particle to complete one full oscillation. (s)
  • Frequency (f): is the number of full oscillations per unit of time. Or the inverse of the period. (Hz).
  • Angular frequency (ω\omega): is the rate at which the state of motion of the particle changes. (rad/s)
  • Wavelength (λ\lambda): It is the distance between two consecutive points in the same phase. It is expressed in m.
  • Wave number (k): It is the number of wavelengths in a distance of 2π2\pi m. It is measured in rad/m.
  • Propagation speed (v): It is the speed at which the disturbance advances through the medium. In m/s
f=1Tω=2πfv=λ⋅fk=2πλ=ωvf = \frac{1}{T} \quad \omega = 2\pi f \quad v = \lambda \cdot f \quad k = \frac{2\pi}{\lambda} = \frac{\omega}{v}
The equation of a harmonic wave, advancing to the right (- sign) or left (+ sign), is given by:
y(x,t)=A⋅sin⁡(ωt±kx+ϕ0)y(x, t) = A \cdot \sin(\omega t \pm k x + \phi_0)
Where ω\omega is the angular frequency, k the wave number and ϕ0\phi_0 the initial phase angle. The time derivative of this expression gives the vibration velocity of each point x with respect to its equilibrium position.
v(x,t)=dydt=A⋅ω⋅cos⁡(ωt±kx+ϕ0)v(x,t) = \frac{dy}{dt} = A \cdot \omega \cdot \cos(\omega t \pm k x + \phi_0)
The second derivative gives us the acceleration:
a(x,t)=dvdt=−A⋅ω2⋅sin⁡(ωt±kx+ϕ0)=−ω2⋅y(x,t)a(x,t) = \frac{dv}{dt} = -A \cdot \omega^2 \cdot \sin(\omega t \pm k x + \phi_0) = -\omega^2 \cdot y(x,t)

Double periodicity of waves

The wave equation is a function in two variables xx and tt, position and time, doubly periodic.
  • If we fix a position in space, x0x_0, we have a resulting function in one variable, time tt of period T:
y(t)=A⋅sin⁡(ωt±kx0)y(t) = A \cdot \sin(\omega t \pm k x_0)
Wave period in time
  • If we fix a specific instant, t0t_0, we have a resulting function in one variable, the position xx:
y(x)=A⋅sin⁡(ωt0±kx)y(x) = A \cdot \sin(\omega t_0 \pm k x)
Wavelength in position
The energy transported by a wave is directly proportional to the square of the frequency and the amplitude:
E=k⋅f2⋅A2E = k \cdot f^2 \cdot A^2
Intensity is defined as the energy that passes through a surface per unit of time in (W/m²):
I=ES⋅t=PSFor a spherical wave I=P4⋅π⋅R2I = \frac{E}{S \cdot t} = \frac{P}{S} \quad \text{For a spherical wave } I = \frac{P}{4 \cdot \pi \cdot R^2}
Spherical wave propagates from a focus F, emits a power P, the intensities at distances R1R_1 and R2R_2 satisfy:
I1⋅R12=I2⋅R22And the amplitudes satisfy:A1⋅R1=A2⋅R2I_1 \cdot R_1^2 = I_2 \cdot R_2^2 \quad \text{And the amplitudes satisfy:} \quad A_1 \cdot R_1 = A_2 \cdot R_2

Intensity in plane and linear waves

Circular waves in the plane. (Ex. wave in a pond). The wave front is formed by a circumference of radius R, and perimeter 2πR2\pi R. The intensity will be:
I=P2⋅π⋅RI = \frac{P}{2 \cdot \pi \cdot R}
The intensities satisfy:
I1⋅R1=I2⋅R2I_1 \cdot R_1 = I_2 \cdot R_2
And the amplitudes satisfy:
A1⋅R1=A2⋅R2A_1 \cdot \sqrt{R_1} = A_2 \cdot \sqrt{R_2}
Linear waves: (ex: propagation in a string). Intensities and amplitudes do not depend on the distance, they are constant:
I1=I2A1=A2I_1 = I_2 \quad \quad A_1 = A_2
It is the phenomenon by which, when a wave strikes the separation surface of two media, it is returned to the first along with part of the energy, and a change in the direction of propagation, according to the following

Snell's Law of reflection:
  • The incident and reflected rays, as well as the normal, lie in the same plane.
  • The angle of incidence θi\theta_i and the angle of reflection θr\theta_r are equal
θi=θr\theta_i = \theta_r
Wave reflection
It is the change of direction of a wave at the separation surface of two media through which it propagates with different speed.

Index of Refraction, n, of a medium is the ratio between the speed of light in a vacuum and the speed of light in that medium:
n=cvn = \frac{c}{v}
Snell's Law of refraction:
  • The incident ray, refracted ray, and the normal all lie in the same plane.
  • The ratio of the sines of the angles of incidence θ1\theta_1 and refraction θ2\theta_2 is equal to the ratio of the propagation speeds
The 2nd law of Snell can take the following forms:
sin⁡θ1sin⁡θ2=v1v2;sin⁡θ1sin⁡θ2=λ1λ2;n1⋅sin⁡θ1=n2⋅sin⁡θ2\frac{\sin \theta_1}{\sin \theta_2} = \frac{v_1}{v_2} ; \quad \frac{\sin \theta_1}{\sin \theta_2} = \frac{\lambda_1}{\lambda_2} ; \quad n_1 \cdot \sin \theta_1 = n_2 \cdot \sin \theta_2
Wave refraction
Every point on a wavefront becomes a point source producing secondary waves of the same velocity and frequency as the initial one. The enveloping surface of the secondary wavefronts constitutes a new wavefront.
Huygens' Principle
It is the deviation in the rectilinear propagation of waves when passing through an opening or passing close to an obstacle.

a) A wavefront reaches an obstacle whose opening is greater than the wavelength. The waves propagate following the rectilinear direction and diffraction is not appreciable.

b) The front reaches an opening comparable to the wavelength λ\lambda. A diffraction effect occurs and they change direction. According to Huygens' principle: in this case, the points of the wavefront in the opening behave as emitting centers of waves whose envelope is the new wavefront. The opening acts as a new wave emitting focus.
Wave diffraction