Wave reflection, refraction and diffraction Knowledge-Based Study

Study notes

1. Behavior of Mechanical Waves at Media Boundaries

   When a mechanical wave travels through a medium, its propagation speed $v$ is determined exclusively by the mechanical properties of that medium, such as density and elastic modulus or tension. The relationship between wave speed, frequency $f$, and wavelength $\lambda$ is given by the wave speed equation:

   $$v = f\lambda$$

   The frequency of a wave is set solely by the vibrating source producing it. When a wave encounters a boundary or an obstacle between two distinct media, the frequency remains constant because oscillations are continuously driven across the interface. Consequently, any change in propagation speed between media directly alters the wavelength.

   • Speed ($v$): Depends entirely on the properties of the transmitting medium.
   • Frequency ($f$): Remains unchanged across all boundaries and medium transitions.
   • Wavelength ($\lambda$): Directly proportional to speed for a fixed frequency ($\lambda = \frac{v}{f}$).

   Reflection of Mechanical Waves

   Reflection occurs when a propagating wave strikes a boundary and redirects back into the original medium without penetrating the second medium. Since the reflected wave continues in the same physical medium, its wave speed $v$, frequency $f$, and wavelength $\lambda$ remain unchanged.

   Reflection behavior follows two fundamental principles:

   1. The Law of Reflection: The angle of incidence equals the angle of reflection ($\theta_i = \theta_r$), where both angles are measured relative to the normal line perpendicular to the boundary.
   2. Boundary Boundary Conditions:
      • Fixed End (Rigid Boundary): When a transverse pulse on a string strikes a rigid boundary, Newton's third law exerts an equal and opposite reaction force back on the string. This reflects the pulse with a $180^\circ$ ($\pi$ radian) phase shift, causing crests to invert into troughs.
      • Free End (Soft Boundary): When the boundary allows free transverse displacement (such as a frictionless ring on a post), the pulse is reflected upright with no phase change.

   Refraction Across Media Interfaces

   Refraction is the change in the direction of propagation of a wave when it passes obliquely from one medium into another where its speed changes. As the wave enters a new medium, its frequency $f$ remains constant, but its speed $v$ changes, which directly changes its wavelength $\lambda$.

   The relationship between the angles of incidence ($\theta_1$) and refraction ($\theta_2$) with the wave speeds in the respective media is expressed by Snell's Law:

   $$\frac{\sin\theta_1}{\sin\theta_2} = \frac{v_1}{v_2} = \frac{\lambda_1}{\lambda_2}$$

   Key consequences of refraction include:

   • Entering a Slower Medium ($v_2 < v_1$): The wave refracts toward the normal line ($\theta_2 < \theta_1$), and the wavelength decreases ($\lambda_2 < \lambda_1$). In water waves, this occurs when transitioning from deep to shallow water.
   • Entering a Faster Medium ($v_2 > v_1$): The wave refracts away from the normal line ($\theta_2 > \theta_1$), and the wavelength increases ($\lambda_2 > \lambda_1$).
   • Normal Incidence ($\theta_1 = 0^\circ$): The wave changes speed and wavelength but does not deviate in direction ($\theta_2 = 0^\circ$).

   Diffraction Around Obstacles and Openings

   Diffraction is the bending and spreading of waves around the edges of an obstacle or through an aperture. Unlike refraction, diffraction occurs within a single uniform medium; therefore, wave speed $v$, frequency $f$, and wavelength $\lambda$ remain strictly constant throughout the process.

   The degree of diffraction depends critically on the ratio of the wavelength $\lambda$ to the width of the opening or obstacle $w$:

   • Significant Diffraction ($w \approx \lambda$ or $w < \lambda$): When the aperture width is comparable to or smaller than the wavelength, the wave fronts emerge as broad circular wave fronts, spreading widely into the geometrical shadow region.
   • Negligible Diffraction ($w \gg \lambda$): When the aperture is much wider than the wavelength, the waves pass through with minimal curvature at the edges, continuing primarily in straight lines with sharp shadows.

   For example, audible sound waves have wavelengths ranging from tens of centimeters to several meters, which are comparable to typical doorway dimensions; hence, sound readily diffracts around corners. In contrast, waves with much smaller wavelengths produce defined shadow zones behind similarly sized barriers.