A Plane Wave Traveling in Medium 1 With R1 = 2.25 Is

Traveling waves

A wave pulse is a disturbance that moves through a medium.

A periodic wave is a periodic disturbance that moves through a medium.  The medium itself goes nowhere.  The individual atoms and molecules in the medium oscillate about their equilibrium position, but their average position does not alter.  As they collaborate with their neighbors, they transfer some of their energy to them.  The neighboring atoms in plow transfer this energy to their neighbors downwardly the line.  In this way the energy is transported throughout the medium, without the transport of any thing.

The blitheness on the correct portrays a medium every bit a series of particles connected past springs.  Every bit one individual particle is disturbed, and so returns to its initial position, it transmits the disturbance to the next interconnected particle.  This disturbance continues to be passed on to the next particle.  The effect is that energy is transported from ane finish of the medium to the other terminate of the medium without the actual transport of matter.  Each particle returns to its original position.

Periodic waves are characterized past a frequency, a wavelength, and past their speed.  The moving ridge frequency f is the oscillation frequency of the individual atoms or molecules.  The period T = 1/f is the time it takes whatever item atom or molecule to go through one complete bicycle of its movement.  The wavelength λ is the distance forth the direction of propagation between two atoms which oscillate in phase.

In a periodic wave, a pulse travels a distance of one wavelength λ in a time equal to i period T.  The speed v of the wave tin exist expressed in terms of these quantities.

5 = λ/T = λf

The relationship v = λf holds truthful for any periodic wave.

If the individual atoms and molecules in the medium motion with simple harmonic motility, the resulting periodic wave has a sinusoidal class.  Nosotros call information technology a harmonic wave or a sinusoidal wave.

Trouble:

A wave on a rope is shown on the right at some time t.  What is the wavelength of this moving ridge?  If the frequency is iv Hz, what is the wave speed?

Solution:

• Reasoning:
  For all periodic waves v = λ/T = λf.
• Details of the calculation:
  The wavelength λ is 3 thousand.  The speed is five = λf = (3 m)(four/s) = 12 m/s.

Problem:

Suppose that a water moving ridge coming into a dock has a speed of 1.5 m/s and a wavelength of 2 thousand.  With what frequency does the wave hit the dock?

Solution:

• Reasoning:
  For all periodic waves v = λ/T = λf.
• Details of the calculation:
  f = v/λ = (ane.five yard/s)/(2 m) = 0.75/s = 0.75 Hz.

Problem:

How many times per minute does a boat bob upward and downwards on bounding main waves that accept a wavelength of forty.0 m and a propagation speed of 5.00 one thousand/due south?

Solution:

• Reasoning:
  For all periodic waves v = λ/T = λf.
• Details of the calculation:
  f = v/λ = (5 m/s)/(40 m) = (0.125/s)*(60 due south/min) = vii.5/min

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Transverse and longitudinal waves

If the deportation of the individual atoms or molecules is perpendicular to the direction the moving ridge is traveling, the wave is called a transverse wave.

If the displacement is parallel to the direction of travel the moving ridge is called a longitudinal wave or a compression wave.

Transverse waves tin occur only in solids, whereas longitudinal waves tin travel in solids, liquids, and gases.  Transverse motion requires that each particle drag with information technology adjacent particles to which it is tightly bound.  In a fluid this is impossible, because adjacent particles can easily slide past each other.  Longitudinal motion simply requires that each particle push button on its neighbors, which tin easily happen in a liquid or gas.  The fact that longitudinal waves originating in an earthquake pass through the center of the world while transverse waves do not is i of the reasons the earth is believed to take a liquid outer cadre.

Link:  Transverse and Longitudinal Wave motility

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Consider a transverse harmonic wave traveling in the positive 10-management.  Harmonic waves are sinusoidal waves.  The deportation y of a particle in the medium is given every bit a role of 10 and t by

y(ten,t) = A sin(kx - ωt + φ)

Hither k is the wave number, k = 2π/λ, and ω = 2π/T = 2πf is the angular frequency of the wave.  φ is called the phase constant.

At a stock-still time t the deportation y varies as a office of position x as A sin(kx) = A sin[(2π/λ)x]
The phase constant φ is determined past the initial conditions of the motility.  If at t = 0 and 10 = 0 the displacement y is zip, then φ = 0 or π.  If at t = 0 and ten = 0 the  deportation has its maximum value, so φ = π/ii.  The quantity kx - ωt + φ is called the phase.

At a stock-still position x the displacement y varies as a office of fourth dimension as A sin(ωt) = A sin[(2π/T)t] with a convenient choice of origin.

For the transverse harmonic moving ridge y(x,t) = Asin(kx - ωt + φ) nosotros may also write

y(x,t) = A sin[(2π/λ)ten - (2πf)t + φ] = A sin[(2π/λ)(x - λft) + φ],
or, using λf = 5  and 2π/λ = k,
y(x,t) = A sin[k(x - vt) + φ].

This moving ridge travels into the positive ten direction.  Allow φ = 0.  Try to follow some point on the wave, for case a crest.  For a crest we e'er have thousand(x - vt) = π/2.  If the time t increases, the position x has to increase, to keep k(x - vt) = π/2.

For a transverse harmonic moving ridge traveling in the negative x-direction we have

y(x,t) = A sin(kx + ωt + φ)= A sin(thou(x + vt) + φ).

For a crest nosotros e'er take chiliad(x + vt) = π/2.  If the time t increases, the position x has to subtract, to keep g(x + vt) = π/two.

Problem:

A traveling moving ridge is described by the equation y(x,t) = (0.003) cos(20 x + 200 t ), where y and ten are measured in meters and t in seconds. What are the aamplitude, frequency, wavelength, speed and direction of travel for this wave?

Solution:

• Reasoning:
  Nosotros have y(x,t) = Asin(kx + ωt), with A = 0.003 m, k = 20 m^-1 and ω = 200 southward^-1.
• Details of the calculation:
  The amplitude is A = 3 mm, the frequency is f = ω/(2π) = 31.83/s, the wavelength is λ = 2π/k = 0.314 thousand, the speed is v = λf = ω/thousand = 10 one thousand/s, and the direction of travel is the negative x direction.

The amplitude A of a wave is the maximum displacement of the private particles from their equilibrium position.  The energy density E/5 (energy per unit book) contained in a wave is proportional to the foursquare of its amplitude.

E/Five is proportional to A²

The power P or energy per unit time delivered past the wave if it is captivated is proportional to the square of its amplitude times its speed.

P is proportional to A²v.

Problem:

To increase ability of a wave by a cistron of 50, by what factor should the amplitude be increased?  (Presume the speed v does not depend on the amplitude.)

Solution:

• Reasoning:
  P is proportional to A^two.
• Details of the calculation:
  P_two/P_one = (A₂/A₁)ⁱⁱ = 50/1.  A_ii = (√50 )*A_ane = 7.07*A_ane.

Link:  The Physics Classroom: Waves

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Interference

Two or more waves traveling in the same medium travel independently and can pass through each other.  In regions where they overlap nosotros but observe a single disturbance.  We observe interference.  When ii or more waves interfere, the resulting displacement is equal to the vector sum of the individual displacements.  If two waves with equal amplitudes overlap inorth phase, i.e. if crest meets crest and trough meets trough, so we observe a resultant moving ridge with twice the aamplitude.  We have constructive interference.  If the ii overlapping waves, still, are completely out of stage, i.due east. if crest meets trough, then the two waves abolish each other out completely.  Nosotros have destructive interference.

Link:  Superposition Principle of Wave  (Delight look at the animations.)