Stationary waves can be produced by the interference of either longitudinal or transverse waves. Stationary waves on strings will be described in terms of harmonics. There are two ways to find these solutions from the solutions above. Hence A = 0. In bounded (finite) medium, the wave arrives at All the particles in the same loop have the same phase at a given instant. The points at which the Some particles vibrate with maximum amplitude and are called antinodes. having same period T. Equation (3) does not represent progressive wave because it does not contain the term tx T . velocities of the two interfering waves are equal and opposite, the resultant When a The spacing between two nearest points is λ/2.

Increasing tension will increase the resonant frequency because wave speed is increased. wavelengths and traveling in the same medium, along the same straight At such points where kx = mπ = mλ/2, sin kx= sin mπ = 0. The formation of stationary waves by two waves of the same frequency travelling in opposite directions. wave does not travel either forward or backward and hence no energy is transported The amplitude R is not constant but varies periodically with the distance x. Also, an antinode is When two, In fact, all the particles in any one segment or loop vibrate in the same phase, while particles in any two adjacent segments are in the opposite phase. If two wave functions ψ1 and ψ2 are solutions, then so is any linear combination of the two: ψ = a ψ 1 + b ψ 2. The points where the amplitude is zero are referred to as nodes. we can conclude that the nodes and antinodes are alternate and equally spaced. displacement is minimum i.e. This means the resonant frequency reduces. unbounded (infinite) medium the wave travels in a given direction continuously At these points ∆y/∆x = maximum, that is strain is maximum. Finally, the example of sound waves in a pipe demonstrates how the same principles can be applied to longitudinal waves with analogous boundary conditions. waves e.g. unbounded medium, has a phase difference of zero radians. This is the equation of stationary wave. Your email address will not be published. Hence this wave can not move forward or backward. The spacing between two such neighboring points is λ/2. interfering waves be represented by the equations. Production of an echo is an example Stationary waves When two progressive waves of similar amplitude, as well as wavelength, travel with a straight line and in the opposite direction which gets superimposed on each other, it leads to the creation of stationary waves. In an unbounded (infinite)) medium, the progressive wave travels in a given direction continuously until it gets dissipated. where the first wave is moving toward the right (positive x), and the second wave is moving toward the left (negative x). amplitude is maximum are called displacement The absence of ± x/λ in the phase term indicates that the wave is traveling neither forwards nor backward. Stationary Waves Page 2 Equation (3) shows that resultant motion is also S.H.M. which require an elastic medium for their propagation is called mechanical The waves The terms fundamental (for first harmonic) and overtone will not be used. The equation of a simple harmonic wave travelling with velocity v = ω/k in a medium is. The wave advancing in the given direction continuously is called a progressive wave e.g. When a The equation of the reflected wave is, Thus, owing to the superposition of the two waves, the resultant displacement at a given point and time is. Equation to find the frequency of the first harmonic. The particles of the medium perform S. H. M. of the same period but the amplitude of the oscillations varies periodically in space. Required fields are marked *, Numerical Problems on Newton’s Law of Cooling. The third special case of solutions to the wave equation is that of standing waves. standing wave or a stationary wave. The distance between the two They are of the same amplitude ‘a’ and have the same period ‘T’ and are traveling with the same velocity ‘v’ along x-axis but in opposite directions. stationary wave passes through a medium, the points of the medium at which the Thus the resultant motion is also S. H. M. of the same period. Increasing the length will reduce the resonant frequency because the wavelength needs to be longer. Analytical equation Suppose the progressive waves of amplitude and wavelength λ travel in the X-axis direction. zero thus amplitude becomes zero are called At these points the strain ∆y/∆x is zero. When they superimpose, the resultant displacement is given by. Explanation of Formation of Stationary Waves: Let the two There are some particles of the medium which do not at all vibrate i.e. Such a wave is called a stationary wave.

They are produced due to interference between the progressive waves which need not be traveling in opposite directions. The distance between a node and the adjacent antinodes is λ/4. Next, two finite length string examples with different boundary conditions demonstrate how the boundary conditions restrict the frequencies that can form standing waves. through the medium. At a given point, amplitude changes with time. The resultant wave travels in the forward direction. The factor of 1/2 is placed for convenience. And there are a number of conditions for which that is the case and those are the conditions which we'll explore. Stationary waves formed on a string and those produced with microwaves and sound waves should be considered.