A series of parallel linear water wave fronts are traveling directly toward the shore at 15.0 cm/s on an otherwise placid lake. A long concrete barrier that runs parallel to the shore at a distance of 3.20 m away has a hole in it. You count the wave crests and observe that 75.0 of them pass by each minute, and you also observe that no waves reach the shore at ±61.3 cm from the point directly opposite the hole, but waves do reach the shore everywhere within this distance. How wide is the hole in the barrier?

Verified step by step guidance

1. Step 1: Start by identifying the given quantities. The wave speed is 15.0 cm/s, the frequency of the waves can be calculated from the observation that 75.0 wave crests pass by each minute (convert this to waves per second), and the distance from the barrier to the shore is 3.20 m. Additionally, the region where no waves reach the shore is ±61.3 cm from the point directly opposite the hole.
2. Step 2: Calculate the wavelength of the waves using the relationship between wave speed, frequency, and wavelength: $v=f\lambda$, where $v$ is the wave speed, $f$ is the frequency, and $\lambda$ is the wavelength. Rearrange to solve for $\lambda$: $\lambda =v/f$.
3. Step 3: Use the concept of diffraction to analyze the problem. The hole in the barrier acts as a single slit, and the waves diffract through it. The angular position of the first minimum in the diffraction pattern is given by the equation for single-slit diffraction: $asin(\theta )\; =m\lambda$, where $a$ is the width of the slit (hole), $\theta$ is the angle to the first minimum, $m$ is the order of the minimum (for the first minimum, $m=\; 1$), and $\lambda$ is the wavelength.
4. Step 4: Determine the angle $\theta$ using the geometry of the problem. The distance from the point directly opposite the hole to the edge of the region where no waves reach the shore is 61.3 cm, and the distance from the barrier to the shore is 3.20 m. Use the tangent function: $tan(\theta )\; =x/L$, where $x$ is the distance to the edge of the region (61.3 cm) and $L$ is the distance from the barrier to the shore (3.20 m). Since $\theta$ is small, you can approximate $sin(\theta )\; \approx \; tan(\theta )$.
5. Step 5: Substitute the values for $\lambda$, $\theta$, and $m$ into the diffraction equation $asin(\theta )\; =m\lambda$ to solve for the width of the hole $a$. Rearrange the equation to find .

Key Concepts

Wave Speed and Frequency

Wave speed is the distance traveled by a wave per unit time, while frequency is the number of wave crests passing a point per second. In this scenario, the wave speed is given as 15.0 cm/s, and the frequency can be calculated from the observation of 75.0 crests passing per minute, which translates to 1.25 crests per second. Understanding the relationship between wave speed, frequency, and wavelength is crucial for analyzing wave behavior.

Diffraction of Waves

Diffraction refers to the bending and spreading of waves when they encounter an obstacle or pass through an opening. In this problem, the waves passing through the hole in the barrier will diffract, creating a pattern of wave fronts that spread out. The width of the hole affects the extent of diffraction, which is essential for determining how far the waves can reach the shore.

Interference Patterns

Interference patterns occur when two or more waves overlap, leading to regions of constructive and destructive interference. In this case, the observation that no waves reach the shore at ±61.3 cm from the point directly opposite the hole suggests a pattern of destructive interference. Understanding how waves interact with each other and the barrier is key to solving for the width of the hole.