Fluids Dynamics - Math Problem Example

1.(ii) Suppose coherent waves with amplitudes A1 and A2 pass through the same point in space. If the waves are in faze at that point – that us, the phase difference is any even integral multiple of π rad- then the two waves consistently reach their maxima at exactly the same instants of time. The superposition of the waves that are in phase with each other is called constructive interference; the amplitude of the combined waves is the sum of the amplitudes of the two individual waves (A1 + A2). Two waves that are 1800 out of phase at a given point have a phase difference of π rad, 3π rad, 5π rad, and so on. The waves are half a cycle apart; when one reaches its maximum, the other reaches its minimum. The superposition of waves that are 1800 out of phase is called destructive interference – the amplitude of the combined waves is the difference of the amplitudes of the two individual waves (|A1 – A2|). For any other fixed phase relationship between the two waves, the superposition has amplitude between A1 + A2 and |A1 – A2|. Suppose two coherent waves start out in phase with each other. Two rods vibrate up and down in step with each other to generate circular waves on the surface of the water. If the two waves travel the same distance to reach a point on the water are different, the phase and interfere constructively. At points where the distances are different, the phase difference is proportional to the path difference. One wavelength of path difference corresponds to a phase difference of 2π radians. So  =  Thus, the phase difference is Phase difference =  x (d1 – d2) = k(d1 – d2) 2(i) The + sign is chosen in the phase because the wave travels to the left. The reflected wave travels to the right, so +kx is replaced wi –kx, and the reflected wave is inverted, so +A is replaced with –A. Then the reflected wave is described by y(x,t) = -A sin ( ωt –kx)] Applying the principle of superposition, the motion of the string is described by y(x,t) = A [sin (ωt +kx) – sin (ωt –kx)] This can be rewritten in a form that shows the motion of the string more clearly. Using the trigonometric identity. Sin α – sin β = 2 cos [ ½ (α +β)] sin [ ½ (α – β)] Where α = ωt + kx and β = ωt – kx We have y( x,t) = 2A cos ωt sin kx Note that t and x are separated. However, in contrast to a travelling harmonic wave, every point reaches its maximum distance from equilibrium simultaneously. In addition, different points move with different amplitudes; the amplitude at any point x is 2A sin kx. (ii) (iii).Figure shows thewave at time intervals of ⅛T, is the period. What you actually see when looking at a standing wave is a blur, with points that never move halfway between points of maximum amplitude. The nodes are the points where sin kx = 0. Since sin n = 0 , the nodes are located at x = n/k = n/2. Thus, the distance between two adjacent nodes is ½ . The antinodes occur where sin kx = ±1, which is exactly halfway between a pair of nodes. So the nodes and antinodes alternate, with one-quarter of a wavelength between a node and the neighboring antinode. So far, we have ignored what happens at the end of the water. The distance between each pair of nodes is ½ λ, so  = L Where L is the length of the traveled and n = 1,2,3,… the possible wavelength for standing waves on water are  The frequencies are  = ( n = 1, 2, 3, …) The lowest frequency standing wave (n = 1) is called the fundamental. Note that the higher frequency standing waves are all integral multiples of the fundamental; the set of standing frequencies makes an evenly spaced set. f1, 2f1, 3f1, 4f1, …, nf1… 3. (i) The frequency of the wave in water is the same as in air. The wavelengths depend on both the frequency and the speed of sound in the medium. The wave length in air is related to the speed of sound in air and the frequency. λair = νairT =  ii) Suppose the motion of the left end of the water is described by y = A cos ωt. By substituting (t x/ν) for t, we obtain the function that describes the motion of any point x > 0. y(x,t) = A cos ω(t-x/ν) To simplify the writing, we introduce a constant called the wave number symbols k, SI unit rad/m k =  =  Then the equation for the harmonic wave can be written y(x,t) = A cos (ωt – kx) The argument of the sine or cosine function, (ωt ± kx), is called the phase of the wave at x and t. Phase is measured in units of angle. The phase of a wave at a given point and instant of time tells us how far along that point is in the repeating pattern of its motion. Since a dine or cosine function repeats every 2π radians, the motions of two different points x1 and x2 that differ in phase by an integer times 2π are exactly the same , the points move “in sync” or in phase with each other. The distance between the two points is an integral number of wavelengths k(x2 –x1)= 2π n (where n is any integer) x2 – x1 =  =  = nλ We try to manipulate the function to see if it can be written as a function of either (t-x/ν) or (t+x/ν) as in the general harmonic wave equation y(x,t) = A cos ω(t-x/ν). The wave speed ν does not appear explicitly in the function as written, but it may be some combination of the other constants in the function. The coefficient of t in our equation should be the constant that represents ω. Factoring out that constant, we have y(x,t) a sin b  = a sin b Now we see that y(x,t) is a function of t + x/ν, where ν = b/c y(x,t) = a sin b where ν =  Therefore, the waves retains its shape since it is a function of (t + x/v); it travels in the –x-direction since the t and x/v terms have the same sign; the wave speed is b/c . Read More