Reflection of Waves
In this section, we explore the behavior of waves when they encounter boundaries. The fundamental principle governing this behavior is that a wave's characteristics can change upon reflection depending on the type of boundary.
Reflection at Boundaries
When a wave approaches a boundary, if the boundary is rigid, such as a solid wall, the pulse or wave experiences a reflection characterized by a phase reversal of π radians (180 degrees). On the other hand, if the boundary is non-rigid, like a string attached to a freely moving ring, the reflected wave retains the same phase as the incident wave.
Mathematical Representation
Mathematically, for an incident wave described as:
$$ y_i(x, t) = a ext{sin}(kx - ext{ω}t) $$
Reflection at a Rigid Boundary results in:
$$ y_r(x, t) = -a ext{sin}(kx - ext{ω}t) $$
Reflection at a Non-Rigid Boundary results in:
$$ y_r(x, t) = a ext{sin}(kx - ext{ω}t) $$
These equations showcase how the amplitude and phase of the reflected wave relate to that of the incident wave, dictating that at a rigid boundary, the maximum displacement becomes negative, indicating a phase change.
Standing Waves and Normal Modes
The section also introduces the concept of standing waves, which arise from the superposition of waves traveling in opposite directions. This phenomenon is observable in systems with two or more boundaries, leading to stationary wave patterns characterized by nodes where displacement is always zero and antinodes where displacement is maximum.
In summary, understanding wave reflection at boundaries is crucial in various applications, including acoustics and structural engineering, and forms a foundational aspect of wave behavior in physics.