Calculate gerstner wave normal, binormal, and tangent by GPU Gems's method.

calculate_gerstner_wave_orientation.hlsl

float3 CalcGerstnerWaveNormal(float3 P)
{
	float3 normal = float3(0, 1, 0);
	[unroll]
	for (int i = 0; i < numWaves; i++)
	{
		Wave wave = waves[i];
		float wi = 2 / wave.waveLength;
		float WA = wi * wave.amplitude;
		float phi = speed * wi;
		float rad = wi * dot(wave.dir, P.xz) + phi * g_time;
		float Qi = steepness / (wave.amplitude * wi * numWaves);
		normal.xz -= wave.dir * WA * cos(rad);
		normal.y -= Qi * WA * sin(rad);
	}
	return normalize(normal);
}

float3 CalcGerstnerWaveBinormal(float3 P)
{
	float3 binormal = float3(1, 0, 0);
	[unroll]
	for (int i = 0; i < numWaves; i++)
	{
		Wave wave = waves[i];
		float wi = 2 / wave.waveLength;
		float WA = wi * wave.amplitude;
		float phi = speed * wi;
		float rad = wi * dot(wave.dir, P.xz) + phi * g_time;
		float Qi = steepness / (wave.amplitude * wi * numWaves);
		binormal.x -= Qi * wave.dir.x * wave.dir.x * WA * sin(rad);
		binormal.z -= Qi * wave.dir.x * wave.dir.y * WA * sin(rad);
		binormal.y += wave.dir.x * WA * cos(rad);
	}
	return normalize(binormal);
}

float3 CalcGerstnerWaveTangent(float3 P)
{
	float3 tangent = float3(0, 0, 1);
	[unroll]
	for (int i = 0; i < numWaves; i++)
	{
		Wave wave = waves[i];
		float wi = 2 / wave.waveLength;
		float WA = wi * wave.amplitude;
		float phi = speed * wi;
		float rad = wi * dot(wave.dir, P.xz) + phi * g_time;
		float Qi = steepness / (wave.amplitude * wi * numWaves);
		tangent.x -= Qi * wave.dir.x * wave.dir.y * WA * sin(rad);
		tangent.z -= Qi * wave.dir.y * wave.dir.y * WA * sin(rad);
		tangent.y += wave.dir.y * WA * cos(rad);
	}
	return normalize(tangent);
}

This video explains how to get the height quite well https://youtu.be/kGEqaX4Y4bQ?si=3XuKA3FUnkIMw1hj&t=802