When it comes to sound reduction, cellulose spray insulation is your best option. Cellulose insulation consists of chemical-treated, recycled newspaper. While this may seem like a flimsy soundproofing option, it actually is quite efficient. It consists of small shavings, so it can easily fill spaces and crevices within the wall, wrapping perfectly around pipes and beams without leaving any open spaces. Second, the insulation packs densely, creating a solid barrier.
Other Insulation Types
Fiberglass insulation has long dominated the insulation market, and it has some benefits when it comes to price, flame resistance and weatherproofing, but it does not make a good soundproofing insulation. The solid consistency of fiberglass actually enables the transference of sound vibrations, potentially making sound problems worse, especially in a small, enclosed location such as a bathroom. Foam insulation is another option; like cellulose, you can find it in a spray variety. Its soundproofing results are comparable to cellulose, though not quite as effective. Cellulose also has the upper hand over foam due to its eco-friendly nature; it's made entirely of recycled newspaper.
Cellulose insulation works on just about any interior wall or roof and offers sound reduction benefits in any of these environments. For bathrooms in particular, you want to use an insulation material that requires minimal destruction of the existing structure so the bathroom isn't rendered unusable by a major renovation project. You can distribute cellulose insulation directly into the wall using an insulation blowing machine. You (or a professional you hire) need only drill a hole between the studs on the wall surface near the ceiling and pump the insulation into the wall. Your drywall can remain intact.
Other Soundproofing Tips
No commercial insulation will fully block the passage of sound waves; you can only hope for sound reduction. To get the greatest sound reduction, you must go beyond insulation. For example, some manufacturers specialize in soundproof drywall. By combining the drywall with the insulation and even adding foam acoustic panels to the wall, you can come close to creating a soundproof bathroom.