Suppose you send an incident wave of specified shape, gI(z-v1 t), down string number 1 . It give

Suppose you send an incident wave of specified shape,...

Suppose you send an incident wave of specified shape, $g_{I}\left(z-v_{1} t\right),$ down string number $1 .$ It gives rise to a reflected wave, $h_{R}\left(z+v_{1} t\right),$ and a transmitted wave, $g_{T}\left(z-v_{2} t\right)$. By imposing the boundary conditions 9.26 and $9.27,$ find $h_{R}$ and $g_{T}$.

Key Concepts

Wave Speed and Impedance Mismatch

Wave speed variations between different media lead to impedance mismatch, which affects the relative amplitudes of the reflected and transmitted waves. The differences in wave speeds are crucial for determining how much of the incident energy is reflected back into the original medium and how much is transmitted into the second medium. This concept is central to understanding the behavior of waves at boundaries.

Boundary Conditions

Boundary conditions are the constraints imposed at the interface between two media, ensuring the continuity of physical quantities like displacement and force (or its related derivative). These conditions are essential for deriving relationships between the amplitudes and forms of the incident, reflected, and transmitted waves, enabling the calculation of reflection and transmission coefficients.

Transmitted Wave

The transmitted wave is the part of the incident wave that continues into the second medium after encountering the boundary. Its behavior is governed by the properties of the new medium, such as its wave propagation speed. The transmitted wave plays a crucial role in energy transfer across different media.

Reflected Wave

The reflected wave is the portion of the incident wave that is sent back into the original medium when it encounters a boundary with a different medium. Its properties, such as amplitude and phase, can differ from those of the incident wave, depending on the boundary conditions at the interface. The reflected wave often travels in the opposite direction to the incident wave.

Incident Wave

An incident wave is the initial disturbance that propagates through a medium. In the context of waves on a string, it is the wave that encounters a boundary or discontinuity between two media and initiates interactions such as reflection and transmission. The incident wave provides the energy and momentum that are subsequently distributed between the reflected and transmitted waves.

Transcript

00:01 Suppose an incident wave is sent of the specified shape given by g z minus v1t downstream number 1.

00:14 It gives rise to a reflected wave given by hr z plus v1t and a transmitted wave given by gtz minus v2t.

00:28 By imposing the boundary conditions, find h .r and g .t.

00:40 The shape of incident wave is given by g .i.

00:46 Z.

00:46 Minus v1t.

00:49 This is the reflected wave, h .r.

00:51 Z plus vanty, and the transmitted wave is gt, z minus v2 t.

00:57 The imposing boundary conditions, which are fz, comma, t, is continued.

01:02 At z equals to 0 and f of 0 t is equals to f of 0 plus t and del f by del z at 0 negative is equal to del f by del z at 0 plus the derivative of f must also be continuous the transmitted wave is the sum of incident and reflected so applying the boundary conditions g .i.

01:36 0 minus v1 t plus hr 0 plus v1 t equals to g t 0 minus v2 t so the equation reduced to g i minus of v1 t plus hr into v1 t equals g t into minus of v2 t applying the boundary condition 2, that is finding the derivative, it will be del of g .i.

02:14 Minus v1t by del t minus del t.

02:20 H .r.

02:21 V1t by del t equals v1 by v2, del g t, v2, integrating on both the sides, the value of integration will be equals to gi minus v1t minus hr v1t equals v1 by v2 gt minus of v2t plus k, where k is the integration constant.

02:59 Adding the equations the first and the second one adding these two equations will give us the value...

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