VIEWS: 99 PAGES: 11 POSTED ON: 2/11/2011
Mechanical Wave Lecture 3:Standing Waves on a String http://physics.ukzn.ac.za/~sonot 1 • Question: What happens to the propagating pulse or sinusoidal wave as it arrives at the end of the string? Incident wave • For a fixed end of a string: At the end, the wave exerts a force on the support-the support exert a force back on the string an sets up a reflected wave. A reflected wave has its amplitude inverted relative to that of the incident wave (see figure (a)) • For a free end of a string: At the end, the string slides along the support and come to rest momentarily, the reflected wave string is now stretched increasing the tension and resulting with the reflected wave. But the amplitude of the reflected wave is not inverted relative to that of the incident. The condition at the end of the string, fixed or free are called the boundary conditions As the wave arrive at the boundary, all or part of the wave is reflected back to propagate in the opposite direction of the incident wave. http://physics.ukzn.ac.za/~sonot 2 • The formation of the reflected pulse is equivalent to the overlap of two pulses travelling in opposite directions. • Consider two pulses with the same shape, one inverted with respect to the other and travelling in opposite direction. • As the pulses overlap and pass each other, the total displacement of the string is the algebraic sum of the displacement at the point in the individual pulse. • Because the two pulses have the same shape and amplitude, the total displacement at point O is zero. • The two pulses, on the right side corresponds to the incident and the reflected pulse in a one fixed boundary situation. http://physics.ukzn.ac.za/~sonot 3 • If the two pulses were not inverted relative to each other, then the total displacement at point O will be twice that of the individual pulse • Slope at point O is zero corresponding to the absence of any transverse force at this point. • Combining the displacement of the separate pulses at each point to obtain the actual displacement is an example of the principle of superposition. • It states that when two waves overlap, the actual displacement at any point on the string at any time is obtained by adding the displacement the point would have if only the first wave was present and the displacement it would have if only the second wave was present. http://physics.ukzn.ac.za/~sonot 4 • Let the wave function of the incident and reflected wave be y1(x, t) and y2(x, t) respectively. The actual resulting wave at the point of overlap is the sum of the individual wave overlapping. • Each wave function, yi(x, t), of wave i must satisfy the wave equation which requires linearity of the wave function. Linear function Nonlinear function • Summation of linear function is also linear. • As a results, if any N linear functions, satisfies the wave equation separately, then their sum also satisfies it and it is therefore a physical possible motion. • Because this principle depends on the linearity of the wave equation and the corresponding linear-combination property of its solution, it is called principle of linear superposition. http://physics.ukzn.ac.za/~sonot 5 Standing waves • Question: What happen when a sinusoidal wave is reflected at the boundary? • The resulting motion when the two waves combine: no longer looks like two waves travelling in opposite direction. • The string appears to be subdivided into a number of segments. • The wave pattern remains in the same position along the string, and its amplitude fluctuates. • Because the wave pattern does not appear to be moving either direction along the string and it is called a standing wave. Recap :Travelling wave: In a wave that travels along the string, the amplitude is constant and the wave pattern moves with speed ν. • The points of the standing wave that never moves at all are called the Nodes (N). • The points midway between the nodes are referred to as antinodes (A). http://physics.ukzn.ac.za/~sonot 6 • Using the principle of superposition, we can now explain how the incident and reflected wave combined to form a standing wave. The red curve shows a wave travelling to the left. The blue curve indicates the wave travelling to the right. Both waves have the same (λ,ν and A) The black curve shows the resulting wave formed by adding the two waves travelling in opposite directions. • At t = 4T/16, the two waves are completely out of phase with each other, and thus the total wave at that instant is zero. • At t = 8T/16, the two wave patterns are exactly in phase with each other, and the resulting wave (black) has twice the amplitude. http://physics.ukzn.ac.za/~sonot 7 • At the nodes, the resulting displacement is always zero. At this point, the displacements of the two waves in red and blue are always equal and opposite and cancel each other resulting with a interface called destructive interference • At the antinodes, the displacements of the two waves in red and blue are always identical giving a large resultant displacement, and this kind of interference is known as the constructive interference. • The wave function of the individual wave are Travelling to the left: Travelling to the right: • The phase of the reflected wave (travelling to the right) is out of phase by 180o relative to that of the incident wave. • Using the principles of linear superposition, the wave function describing the resulting wave is given as: http://physics.ukzn.ac.za/~sonot 8 Standing wave on a string: • Shows that at each instant, the shape of the • The wave shape stays in the same string is a sine curve position, oscillating up and down • At the nodes, sin(kx) = 0 for the displacement to be always zero, hence • A standing wave, unlike a travelling wave, does not transfer energy from one and to the other. The two wave that forms it would individually carry equal amounts of power in opposite directions. There is a local flow of energy from each node to the adjacent antinode and back but the average rate of energy transfer is zero at every point. http://physics.ukzn.ac.za/~sonot 9 • Question: What happen if the string has a definite length L, rigidly held at both ends? .g. string of a guitars and violins • The standing wave that results must have a node at both ends of the string. Since the nodes are always (λ/2) apart, the length (L) of the string must be: For the standing wave to exist. The standing-wave wavelength is given as • Note: if l=2L/n, individual waves can still exist but their summations cannot be a standing waves • The standing wave frequencies is therefore: http://physics.ukzn.ac.za/~sonot 10 • Where f1 is the smallest frequency corresponding to the largest wavelength (n = 1), • f1 is called the fundamental mode. • The higher frequency are called harmonics or in musician language are called overtones. n Harmonics Overtones 2 2nd 1st 3 3rd 2nd n nth (n -1)’s overtones • Each of the wavelength given by, λn, for a string fixed at both end, corresponds to a possible normal modes pattern and frequency. • Normal modes of an oscillating system is a motion in which all particles of the system move sinosoidally with the same frequency. http://physics.ukzn.ac.za/~sonot 11
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