How does a husband encourage his wife who has Alzheimer’s to continue bathing?

I hope that the wife can see a Certified Health Aide coming in to assist her as a "beauty treat". Getting her "hair done" starts with a thorough shampoo, and then a blow dry. I think the wife will cooperate better with an Outsider coming in. The wife could benefit from a personal care assistant for bathing and personal hygiene, most certainly. Would the wife still be able to go to the Beauty Shop/Hair Salon, where she would be leaned back in comfy chair, hair washed into a basin? If that would frighten her, then it's a bad idea. I can see from your profile that you are a helper and advocate for the elderly and infirm. You most likely have the skill to assist the husband in making a big deal about the "Beauty Day" that occurs when the personal care assistant/CNA comes in to bathe wife, clip her nails (maybe with a hand massage), get a good thorough shampoo, etc. Most husbands might be clueless as to how to pull this off (Beauty Day) but you "Make a Difference" have a magic touch with the elderly and you can help husband set the stage for this to happen. He's not going to be able to encourage her enough to get her to take charge of her personal hygiene. A big Tea Party with Beauty Day is much more likely to appeal to the husband and the wife, with a Personal Attendant coming in to "pamper" his wife on her Beauty Day. Worth a try, anyway.

Visiting Angels

I agree that you should try (if you haven’t already) having an aide come in to shower her. We cared for my MIL when she was on hospice and before hospice was set up I showered her several times. I could tell she was much more comfortable once she had an experienced aide helping her (she didn’t have dementia). I did my best but I just don’t have the experience and confidence that she did. If she has any aversion to bathing she may try harder to manipulate someone closer to them than they would a stranger. Also it’s very important to have the room warm enough and a comfortable, secure seat for her.

I got a Certified Nurses assistant who helped with bathing and Hygiene . They order a special chair to slide them On in the shower . Medicare covers the cost. At the doctors we had a Podiatrist . The CNA was terrific and she became friends with My Dad and did Light housekeeping . You May want to Lighten your Load and get some help with Hygiene and Bathing .

In addition to Ohwow's suggestion - taking her out to the salon for a shampoo a couple of time a month might work for her hair (depending on her previous habits). People who have bought bidet toilet seats rave about how fresh and clean they feel.

There's no sure way of doing that as each brain and each person with this disease is different. Approaches are "nested" according to the situation and the family/people and the surrounding circumstances about the scenario. You may roll your eyes at this answer, but it's true.
I've seen people with advanced dementia/Alzheimer's sit perfectly still getting bathed, appearing more than cooperative. And then I've seen people who were just diagnosed become very combative and angry when others try to help.
Perhaps if the person you're speaking about doesn't want to bath or lacks the know how to do so, it may take further evaluation and a softer approach or an approach that is more assisted.
Clearly, the husband is in need of some homecare assistance with caring for his wife or it's time to move her into memory care/facility.

Make sure that she is comfortable. The water is at the right temperature, grab bars installed, etc.

Do sponge baths in between if needed. Also, dry shampoo is a help.

Do you have outside help? People seem to cooperate more with others. They may be able to convince her to do things more easily.

Wishing you and your wife well.

You may need to have someone come in and assist. Like visiting angels or another care giving service. Sometimes an outsider can get the job done whereas a family member may be easily intimidated by the crying or words used by the patient. Blessings to you this is not easy